Well, I think it's great that your interested in learning about your water chemistry. Especially if your planning on cycling your new tank with fish. There is also a lot of info on the internet about fishless cycling if you want to go that route.
You'll want test kits for ammonia, nitrite, pH, and nitrate. There are plenty of others, but those are the most important from the fishes perspective. I have kits from Tetra and Salifert. I think I spent around $20 for the Tetra Master Kit and maybe $7 each for the Salifert kits. The Tetra Master Kit does not include a nitrate kit, so you'd have to buy that separately. A lot of people seem to be happy with the AP kits, too. Stay away from any kits that use test strips. You want either liquid of powder reagents because they are more accurate. Even so, none of commonly available kits will be perfect, but they will give you a pretty good result.
As for how you do the testing, the kits come with directions, which you'll need to follow precisely for accurate results. One nice thing about the Tetra kit, it includes a fact sheet on each test as to why you're doing it and what do with the results when you get them. You can also get all that info off the internet.
On your existing tank, you'll get the most use out of the nitrate test. The ammonia and nitrite will only be useful if you start having problems. It's always good to know the pH of your water and that of the tank your buying fish from for acclimating purposes.
Hope that helps.