Okay, what I would do in your case is start conducting daily water changes for the next while. It's awesome that you have that test kit-- it's a good one, and you will certainly need it for the next while! Your tank is going through a cycle to develop nitrifying bacteria. Understanding how this cycle works and how it progresses will help you to grasp what's going on in your tank, what those readings you're getting actually mean, and will form a large part of the foundation of knowledge for making you a wonderful fishkeeper. 
It sounds like you already have an idea of what cycling a tank means. To help you a little more, here's a good article for you to read to get a better idea of what it entails (and here's another from right here on AC). Basically, your tank's bacteria population at the moment is way overrloaded-- there's just too much fish waste in there for what little bacteria you have to deal with. That's probably why your ammonia and nitrite readings are so high. Your bacteria colony will catch up eventually, but it wll take time, and you'll have to do as much as you can to keep your fish healthy and comfortable during this transition process.
Now, ready your test kit! You're about to begin a mission: your goal is to get your ammonia and nitrite levels down to 0, and to maintain your nitrate levels at or below 20, and keep those readings steady. Your pH is actually probably good where it is, so we won't worry about that.
Here are some things you can/should do while getting your tank cycled:
1) First and foremost, I would recommend doing daily water changes, as I said before. This will help dilute the amount of ammonia/nitrite/nitrate already in the tank, and is very important right now. High levels of ammonia can cause burns on the gills of your fish, which are irreparable and will impair them for life. I'd say you can do anywhere between 30% - 50% water changes everyday without causing any harm-- some people here may even suggest doing larger ones. Make sure you use a waer conditioner that gets rid of chlorine/chloride if you're on city water! These chemicals are added to municipal water supplies to kill bacteria, and will really be counterproductive in cultivating your good bio-bugs in your filter.
2) Limit the amount of food you're giving to your fish. Although they may try and tell you they're starving, your fish really don't need to be fed every day. More food will mean more things in your water that can decompose and produce more pollutants, either as uneaten food that rots on the floor, or as more poop from your fish. Feed them sparingly once every other day at the most for now until your levels calm down. They'll be fine.
3) "Seed" your tank to help kick start the population of bacteria. Seeding involves borrowing an amount of bacteria from an established tank and adding it to your own-- since nitrifying bacteria are not fond in great numbers within the actual water itself, adding water from your pond or from someone else's tank won't help much. Instead, if you know someone who has a tank that is up and running, and has been for a long time, see if you can snag some of their filter media, or gravel. Filter media is best to borrow, since it can be placed into your own filter, and usually contains the most bacteria. Gravel will help, too, though maybe not as much-- either add the gravel straight on top of your own substrate, or stuff it in a media bag and let it just hang out in the back of the tank, to be removed when your tank is stable.
Another method of seeding your tank is to get some plant wool. Go to your pet store, and see if you can find any potted aquatic plants for sale-- usually, these plants have their roots wrapped in a wooly substance before being stuffed in plastic pots. I think this stuff is called rock wool? Anyway, it's usually loaded with nitrifying bacteria. Get a cheap plant or two, pick off the wool, put it in a media bag and either add it to your filter, or let it sit in the tank like you would for the gravel. What you do with the plant is up to you, although some may suggest keeping it, because...
4) Plants use the very same nutrients we are trying to get under control as part of their biological activity, and are as such sometime suggested as good addition to new tanks. However, caution should be exercised in doing so-- plants will only help if they are kept healthy, and will likely only make a noticable dent if they are of a fast growing species. Keeping live plants in aquaria can be very rewarding, but requires special considerations-- if this is something you might want to consider, by all means, check out the plants forum here on AC. There, you will find many topics about newbie-friendly plants, and lots of knowledgable brains to pick that will be more than happy to make suggestions as to which plants will work for you. Remember though, that keeping plants will only help you in your cycle if you can keep them from dying-- a dead plant is just another source of pollutants. So please do ask before persuing this course, and we'll steer you in the right direction.
5) Using the right additives will help accelerate cycle, and may help alleviate the stress on your fish. If you live in the United States, you may be able to get your paws on a product called BioSpira. Unfortunately, I live in Canada, and am unable to purchase it myself, so I have no experience with it at all. However, it has been used by many people here with great success. This product apparently contains live bacteria that will jump start your cycle in much the same way as seeding will-- if you can find it, make sure it has been kept in a refigerator, and that it is a relatively fresh batch. This stuff can be hit or miss, but when it works, it does wonders. Be wary that there are plenty of other products on the market that will claim to do similar things, like Cycle and BioZyme, for example, but these products don't have nearly the same success rate, or may simply contain the wrong kinds of bacteria altogether. BioSpira on the other hand is known to actually work, and so is the only one I would recommend you try.
I'm not familiar with the AquaZyme you've been using. It sounds like Amquel, though, which is a product that binds to ammonia and converts it to a less toxic substance. If this is the case, be aware that using it may mess with your ammonia readings in your test kit, and give you deceptively high values. It may also stall your cycling process, depending on whether or not the bacteria can stll process the locked ammonia as they would normal toxic ammonia. I wish I had more experience with the stuff you're using to tell you for sure... Try maybe running a search on the forum to see if anyone else has said anything about it?
Phew! Sorry about the wall of text.
For the time being, that's about all the advice I can think of giving you. I'm sure others will chime in with more sage words to fill in any gaps I've left.
No matter what you do to help get your cycle going, make sure you do those daily water changes! Keep checking your levels for ammonia, nitrite and nitrate every day, and keep a record of them on a memo pad or something. That way, you can keep track of how your levels are changing day to day, which will show you how your cycle is progressing, and feel free to keep us posted! Good luck!
It sounds like you already have an idea of what cycling a tank means. To help you a little more, here's a good article for you to read to get a better idea of what it entails (and here's another from right here on AC). Basically, your tank's bacteria population at the moment is way overrloaded-- there's just too much fish waste in there for what little bacteria you have to deal with. That's probably why your ammonia and nitrite readings are so high. Your bacteria colony will catch up eventually, but it wll take time, and you'll have to do as much as you can to keep your fish healthy and comfortable during this transition process.
Now, ready your test kit! You're about to begin a mission: your goal is to get your ammonia and nitrite levels down to 0, and to maintain your nitrate levels at or below 20, and keep those readings steady. Your pH is actually probably good where it is, so we won't worry about that.
Here are some things you can/should do while getting your tank cycled:
1) First and foremost, I would recommend doing daily water changes, as I said before. This will help dilute the amount of ammonia/nitrite/nitrate already in the tank, and is very important right now. High levels of ammonia can cause burns on the gills of your fish, which are irreparable and will impair them for life. I'd say you can do anywhere between 30% - 50% water changes everyday without causing any harm-- some people here may even suggest doing larger ones. Make sure you use a waer conditioner that gets rid of chlorine/chloride if you're on city water! These chemicals are added to municipal water supplies to kill bacteria, and will really be counterproductive in cultivating your good bio-bugs in your filter.
2) Limit the amount of food you're giving to your fish. Although they may try and tell you they're starving, your fish really don't need to be fed every day. More food will mean more things in your water that can decompose and produce more pollutants, either as uneaten food that rots on the floor, or as more poop from your fish. Feed them sparingly once every other day at the most for now until your levels calm down. They'll be fine.
3) "Seed" your tank to help kick start the population of bacteria. Seeding involves borrowing an amount of bacteria from an established tank and adding it to your own-- since nitrifying bacteria are not fond in great numbers within the actual water itself, adding water from your pond or from someone else's tank won't help much. Instead, if you know someone who has a tank that is up and running, and has been for a long time, see if you can snag some of their filter media, or gravel. Filter media is best to borrow, since it can be placed into your own filter, and usually contains the most bacteria. Gravel will help, too, though maybe not as much-- either add the gravel straight on top of your own substrate, or stuff it in a media bag and let it just hang out in the back of the tank, to be removed when your tank is stable.
Another method of seeding your tank is to get some plant wool. Go to your pet store, and see if you can find any potted aquatic plants for sale-- usually, these plants have their roots wrapped in a wooly substance before being stuffed in plastic pots. I think this stuff is called rock wool? Anyway, it's usually loaded with nitrifying bacteria. Get a cheap plant or two, pick off the wool, put it in a media bag and either add it to your filter, or let it sit in the tank like you would for the gravel. What you do with the plant is up to you, although some may suggest keeping it, because...
4) Plants use the very same nutrients we are trying to get under control as part of their biological activity, and are as such sometime suggested as good addition to new tanks. However, caution should be exercised in doing so-- plants will only help if they are kept healthy, and will likely only make a noticable dent if they are of a fast growing species. Keeping live plants in aquaria can be very rewarding, but requires special considerations-- if this is something you might want to consider, by all means, check out the plants forum here on AC. There, you will find many topics about newbie-friendly plants, and lots of knowledgable brains to pick that will be more than happy to make suggestions as to which plants will work for you. Remember though, that keeping plants will only help you in your cycle if you can keep them from dying-- a dead plant is just another source of pollutants. So please do ask before persuing this course, and we'll steer you in the right direction.
5) Using the right additives will help accelerate cycle, and may help alleviate the stress on your fish. If you live in the United States, you may be able to get your paws on a product called BioSpira. Unfortunately, I live in Canada, and am unable to purchase it myself, so I have no experience with it at all. However, it has been used by many people here with great success. This product apparently contains live bacteria that will jump start your cycle in much the same way as seeding will-- if you can find it, make sure it has been kept in a refigerator, and that it is a relatively fresh batch. This stuff can be hit or miss, but when it works, it does wonders. Be wary that there are plenty of other products on the market that will claim to do similar things, like Cycle and BioZyme, for example, but these products don't have nearly the same success rate, or may simply contain the wrong kinds of bacteria altogether. BioSpira on the other hand is known to actually work, and so is the only one I would recommend you try.
I'm not familiar with the AquaZyme you've been using. It sounds like Amquel, though, which is a product that binds to ammonia and converts it to a less toxic substance. If this is the case, be aware that using it may mess with your ammonia readings in your test kit, and give you deceptively high values. It may also stall your cycling process, depending on whether or not the bacteria can stll process the locked ammonia as they would normal toxic ammonia. I wish I had more experience with the stuff you're using to tell you for sure... Try maybe running a search on the forum to see if anyone else has said anything about it?
Phew! Sorry about the wall of text.
No matter what you do to help get your cycle going, make sure you do those daily water changes! Keep checking your levels for ammonia, nitrite and nitrate every day, and keep a record of them on a memo pad or something. That way, you can keep track of how your levels are changing day to day, which will show you how your cycle is progressing, and feel free to keep us posted! Good luck!
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