The number 1 chemical you will need is that water conditioner for the tap water. However, which one you need depends on your tap water. The only way to know about your tap water is to test it, so the very first thing to get is the test kit and that API kit is the one a great deal of us (myself included) will recommend. Now test your pH of the water directly out of the tap as well as test to see if you have an ammonia reading out of the tap. If you have an ammonia reading then you most likely have chloramines rather then just plain chlorine in your water. If you do not have any ammonia then it's a better chance that it's just plain chlorine. So if it's plain chlorine then something along the lines of
AquaSafe,
NovAqua+, or API's
Tap Water Conditioner. It will make the tap water safe to add to the tank. However, none of these will do anything to protect against ammonia or nitrite. If you have an ammonia reading from the tap then you will need to go the step farther and deal with it with a product like
Prime (what I use), or
Amquel+. If you are just starting to cycle your tank, then you might want to get one of those anyway because even if you don't have ammonia in your tap water, you will have some in your tank shortly. Prime and Amquel+ will make it safe... not remove it, but make it safe for the fish and still allow the good bacteria to convert the ammonia to nitrite and the nitrite to nitrate. Now, beyond that, if you have a good pet store close by then you don't really "need" anything else at this point. Once you have a problem you can get the right chemical for the job. Yes they keep for a while but not forever. So as long as you have almost instant access to it via a nearby pet store, let them store it and save your money for other things.
As for the pH test I mentioned, you don't need any chemicals for that, but it's good to be sure that your tank matches whatever reading you get from the tap. If it's the same or very close then you are good to go. If it drastically different then we need to find out why because it could cause problems over time. Good Luck!