I got my first fish about 5 or 6 years ago. I chose a Betta fish because they seemed less complicated and they were very beautiful.
I bought a fish because at the time I was living in the dorm in college, and they were the only pets allowed. After the long winter, I was struggling with a bit of depression, and I really, really missed having an animal around to care for. Getting a fish was the best thing for me - having the life, beautifully colored, really raised my spirits.
Betta fish are really easy fish for the beginner, I thought. They are also known as Siamese Fighting Fish. Most of the ones you can buy in a store are males, because they are the pretty ones with beautiful, flowing fins. Females have short fins and aren't usually as vividly colored. They have to be kept by themselves, as they will attack and eat other fish. (Even breeders, I read, only put the two fish together long enough for breeding and egg-laying, before separating them again!) They do fine by themselves, and you can entertain them by holding up a mirror and watching them puff out their gills and fins ferociously because they think they see another fish. :-)
When you're getting a fish for the first time, you'll need to first consider how you're going to house them. Generally, the rule of thumb is a gallon for each inch long the fish is. (It really depends on the fish, though - some fish require more space than others.) Bettas are nice because they can survive in a smaller space than most fish. A lot of people just put their betta in a bowl or vase.
It's been a while since I researched this so I'm going from memory here, but I would actually caution against just using a bowl for your betta (surprised?). Getting a tank with a filter will keep the water fresher/cleaner, and increase the time between water changes, which reduces the stress on the fish. Also, I found that Zeke was very active and responded to me, compared to other fish kept in a bowl who just sat there unmoving.
I bought an aquarium kit, like the one here. I really liked this particular kit. I had the two-gallon one, but I think a one-gallon tank would do just fine as well. It comes with an air stone filter, which just keeps air bubbles circulating through the tank (rather than collecting and cleaning the water). It also has a hood with a light, and the best part - it is completely cat-proof (unless your cat is particularly adept). Bettas also jump, so some sort of lid is essential - I used to put Zeke in a bowl when I was changing his water, and occasionally he would jump out of it.
You'll need aquarium stones for the bottom of your tank - they're easy to find in pet stores, or in the pet section of stores like Walmart. You'll want enough stones to fill a few inches on the bottom of your tank. You will also need plants (as many as you have room for). Bettas' natural habitat are the rice patties of Thailand, and so they are used to having a lot of places to hide. Therefore, you'll want to make sure their tank has plenty of plants, without being two crowded (of course). I had three plastic plants that seemed to serve my Zekes very well. Other than that, you should be set - bettas don't need much in the way of toys. :-)
The food is pretty simple - you can buy Betta Bits from the store. You can also get treats for your betta, which they love - freeze-dried bloodworms. Zeke used to get so excited when I gave him some bloodworms! There are probably other variations of the food, but to be honest I never researched it. :-)
Next I'll talk about setting up the tank for your new betta.