-What tank size do you recommend?
Much like with freshwater, bigger is generally considered easier to keep stable. Though, overstocking is even more of an issue in saltwater than fresh, so you'll want to make sure to stock pretty light, especially at first.
50g+ would be a good size.
-What kind of filter should I get?
That is a hard question to answer. In a fish only tank you have many of the same options as freshwater. Overflow w/ sump, hang on back (though you get salt creep buildup with this), canister. Anything that will filter out ammonia and nitrites. There are many hardy marine fish that do just fine as long as you filter the tank and keep nitrates fairly low.
The big problem with only using mechanical filters for all your filtration is that you end up with a buildup of nutrients, moreso than using liverock (which will filter nitrates also), therefore you might have a lot of algae growth, which is ugly, though not necessarily bad for the fish.
In a reef tank your choices will be more limited depending on what you keep and what kind of maintenance you want to do.
You can use just live rock for your filtration and powerheads for water turnover (I've kept this setup just fine). You can use a combo of liverock and mechanical filters (either sump, canister, hang on back, etc., but you will want to clean the media of gunk fairly often to keep the nitrates down)
You can use JUST mechanical filters but you will have more trouble keeping your parameters stable enough to keep the tank looking good, and some corals and inverts are very picky about water quality.
You can use a sump with liverock in it for filtration.
But IMO, lots of live rock, no matter what other filters you use, is the best way to keep the tank stable. You can also add live rock to a sump.
Live rock will need to be cured (left in a tank for a certain amount of time until the die off is complete. Any live rock will have die off when you move it, and how much it has will depend on how much life was on it before you brought it home, how far you moved it and how long it was out of the water. This is also one way to cycle a saltwater aquarium. The die off will cause ammonia, which will be converted to nitrites by bacteria and then to nitrates just like in a freshwater aquarium. It does not stop there in saltwater though as there are anaerobic bacteria within the rock that will also filter nitrates. So once the die off is complete and your ammonia and nitrites are gone, and your nitrates are dropping or have disappeared, your cycle is complete).
You do not want to add a large amount of live rock to an established tank because it can cause an ammonia spike from die off. You do not want to add stock to a saltwater tank with brand new live rock because the die off is not complete (most likely) and your organisms will be stressed or die during the ammonia and nitrite spike.
-Do I need a protein skimmer?
No you don't. But it will help. A good protein skimmer will keep your tank from having as much organic buildup, which leads to lower nitrates and a more stable system.
That said I have run reef tanks and fowlr without protein skimmers.
Protein skimmers will help A LOT with predator tanks though. Carnivorous fish like eels will produce a ton of waste that a protein skimmer is very good at pulling out.
If you don't have the funds for a protein skimmer right off the bat though there's no reason you can't run a tank without one.
-What type of lighting?
That's completely dependent on the type of tank. Corals will need specific lighting depending on the species, and the amount/spectrum is going to depend on the coral species and the depth of your tank.
For a fish only tank or fowlr you can run any light that makes the tank look good for viewing. You just don't want to run too much light in a fish only or fowlr because it will cause pest algae growth.
-How many fish can I put into the gallons that you recommended?
That's strictly dependent on species size, aggression, their sensitivity to organic waste, how much maintenance you plan on doing, your filtration setup, etc. ...
You need to determine all of that before deciding how many fish.
-What would be the total estimated cost of getting saltwater aquarium with setup (not including the fish)? (My old aquarium is cracked so I will need to get a new one.)
Between $1 and several thousand lol.
Seriously depends on what brand of equipment you buy, what type, what setup you're going for (reef will need to include lighting in the cost).
You can setup a fish only tank for the same cost as a freshwater aquarium, only with the added cost of salt.
If you want to do fowlr or reef you will need liverock, and it varies between $1.50 lb, and $8.00 lb on average, depending on where you get it and the quality of the live rock. Live rock that is very porous, therefore weighs less per size, looks attractive, etc., will be more expensive, but you may need less. I wouldn't suggest getting the extremely cheap live rock as it tends to be heavy and non porous, therefore doesn't support as many bacteria or as much life.
Say you pay $5.00 lb for liverock (you can usually get decent quality for that), and for a fowlr you'll want at least 1-2 lbs per gallons. For a 50g tank that would be $250-$500 for your tank.
For a fish only tank you only need as much rock as you find attractive, or none at all.
You'll need heaters which will be the same cost as in any other aquarium.
You'll need salt, and how much you use will be partially determined by what salinity you will keep (reefs need more specific salinity. in fish only you can often get away with less salt). Prices of salt varies depending on brand and many people have preferences depending on what they keep (reefkeepers are usually more picky because some brands of salt are inconsistent, therefore won't buffer the water properly, etc.).
A 5g bucket of instant ocean, which is usually decent in general, is $30-$50. I used to use a mix of instant ocean and oceanic for my reef tanks just because it seemed to buffer the water better (therefore have to add less buffering solution) and seemed to come out closer to the correct amount of calcium for my tank.
You'll need a refractometer (recommended personally) or a hydrometer. Hydrometers are 5-10 bucks, but I find them to be inaccurate and each one seems to measure different. My refractometer was $50, but I don't have to replace it (the hydrometers seem to break or get stuck often) and it works great.
As for a tank obviously you'll just have to shop around for the best price.
Lights will vary DRASTICALLY by price depending on brand, type, power, etc.. so you'll need to look into those when you decide what kind of setup you will need for what you're stocking.
You will need chemical test kits. For fish only or fowlr, you may only need ammonia, nitrite, nitrate and pH (pH is more important in a saltwater setup than freshwater, so you can't go without it).
For a reef or a tank with inverts you'll want to also monitor the level of calcium, gH, kH, and possibly phosphates.
An API master kit for saltwater is ok in general, but the calcium test especially is not accurate, so if you're doing inverts or corals you may want to look for a more accurate test.
You'll probably want powerheads, which can vary from the simple ones you use in your freshwater tanks, to some very expensive ones that move the water a specific way. But for a first saltwater tank normal powerheads are usually fine.
Sorry I can't be mores specific. There's just SO MANY ways you can keep a saltwater tank and so much money you do or don't have to spend depending on that.
