If you want "little fishies" then livebearers (guppies, mollies, platties, swordtails) are probably your best bet. They don't lay eggs, but give birth to live young and will breed without much/any prompting. That said, if you're not set up for it, you may find that you will lose most of the fry. The other thing is, if you're looking to breed fish, you'll want to have somewhere to put them as they grow. There's not really room for any more fish in the tank that you have right now, so if your fish were to spawn, you'd want to have somewhere for the additional fish to go.
There are lots of different water conditioners on the market. I know a lot of folks here like Prime. Just make sure that whatever you get will deal with both chlorine and chloramines as there's a good chance that you'll have one or both of those in your water if you're on a municipal water supply.
A test kit is a good idea so that you can see what your water parameters are. The most common recommendation that I see on the boards here is the API Master Test Kit. This is a liquid kit with tests for ammonia, nitrites, nitrates, and pH (two pH tests, one for low and one for high). Other important water parameters are GH and KH, but again, if you're on a municipal water supply you can usually get that information from the township/city without having to test it yourself. You can also get test strips that will give results for some combination of the things mentioned above. The strips are sometimes cheaper, but often less accurate.
As far as water parameters go, ideally what you're looking for is ammonia = 0, nitrites = 0, nitrates <20ppm. Basically, fish produce ammonia. Ammonia is very toxic. There are bacteria that turn to ammonia into nitrites. Nitrites are less toxic than ammonia, but still not something you'd want to be swimming around in and breathing. There are other bacteria that turn the nitrites into nitrates. Nitrates are much less toxic. However, if you get enough of them in the water, they can make your fish sick. When you take out 20% of the water with your gravel vacuum, you're also taking out 20% of the nitrates. When you add fresh water back in, you're diluting the nitrates that are left. Once you get a test kit, you can try testing for nitrates before and after a water change to see how it affects the nitrate concentrations.
The beneficial bacteria that I mentioned above will grow on most of the surfaces in your tank. The majority of them will be in your substrate (gravel) and your filter. That's part of how your filter helps keep the water healthy, it gives all those bacteria a place to grow and turn more harmful compounds (ammonia and nitrites) into less harmful ones (nitrates). A lot of bacteria live in the white, spongy part of your filter. It's actually okay not to replace it, since taking away that sponge will also take away a lot of your bacteria. What you should do every once in a while though is try to clean/rinse the sponge so that it doesn't get clogged up. You can do this when you are doing a water change. Just rinse your filter sponge in the old tank water before you throw it out. Don't rinse it in regular tap water since the chlorine will kill your beneficial bacteria.
Also, if you don't already have one, you might want to pick up a thermometer. You should be able to buy a cheap, stick-on one for a couple of dollars or a glass one to go inside the tank for about a fiver. This way you can make sure that your heater is actually keeping the water at the temperature you want it at.
@austinwend: It may depend on where you are located, but pretty much any fish or pet store in my area will sell a "10 gallon starter kit". This normally comes with a 10 gallon tank and hood as pictured above. Usually with an incandescent light of some sort. A small filter. And often some samples of food, water conditioner, and various other products. You usually have to buy the substrate (gravel or sand), heater (if you're planning on tropical fish), thermometer, water test kit, and any decor separately.
Welcome to the hobby and I hope you continue to enjoy your fish!