Let's see, here's a basic list of things:
1) 2.5 gallon tank, minimum. Unless you want to do 50% water changes every day. In my opinion, a 0.5 gallon tank isn't big enough for any fish, small or not.
2) Thermometer. Betta fish are tropical fish, they need water temperature that's in the 76-82 degree range
3) A heater, preferably adjustable, if it's not hot enough to maintain that temperature where you live
4) Water conditioner, which you already have. A water conditioner called Seachem Prime is the kind many members here recommend, it keeps ammonia and nitrites from being toxic to your fish
5) Cover in the tank. Betta fish like a lot of places to hide. Many members here recommend putting a few, or a lot of silk plants or live plants, plastic plants can tear their fins.
6) A test kit. You have to monitor water conditions. In small bowls or tanks, ammonia builds up quickly and will slowly poison your fish, and cause many health issues. Test strips are okay, but a liquid test kit called the API Master Test kit is recommended on these forums. Ones that will test ammonia, nitrites, and nitrates are fine.
7) Water changes! In that tiny bowl, as I said, you need to change the water every day. Make sure you match the temperature of the water you're putting in to the water you take out, if you don't, you risk putting your fish into temperature shock. That's why a thermometer is very important.
8) Research! I recommend you look up information on the nitrogen cycle in fish tanks. It's very helpful, and it gives you a lot of information on caring for fish. Here's a link, it's a good read.
http://www.bettafish.com/showthread.php?t=507585
This is also good for information on Betta care, please give this a read when you can, too!
http://www.bettafish.com/showthread.php?t=232570
Good luck with your fish friend! A word of advice, since a see a lot of mistakes happen this way.. Don't listen to pet store people, do your own research. I'm glad you came here asking for help. Everything may seem a bit complicated and daunting at first, it was for me too, but it's very rewarding once you get the hang of things.
Any other members feel free to add anything I missed, or correct anything!