Yes, you can leave your betta in the bowl while you cycle, doing plenty of water changes every other day. Fish-out cycle can be safer, but only if the fish is getting plenty of water changes in his current container. Cycling the tank isn't always necessary with bettas, but it's a good idea, and helps prevent fin rot from ammonia build up. While he's in is bowl, you don't actually need a heater then, unless you live in a hot place, where temperature fluctuations are pretty major. You need a heater and a thermometer for is cycled home. That is very important. And filter's aren't needed for success, but if you aren't ready do do the number of water changes that that set-up requires, that it's not a possibility, get a filter. And if it's not to lat, and you haven't gone to the store yet, DON'T GO NEAR THE BETTA TANKS! That is, unless you ant to get another betta against your better judgement. You may think that that's crazy, but that's what everyone here first thought. Now many of us have do do shopping for supplies online, just to avoid temptation.
For food- most betta pellets are okay, just make sure it doesn't have a plant product in the first few ingredients. And if you can, buy some of the betta treats like freeze dried blood worms, or shrimp. Some members here have had problems with those when fed too often, so only buy as a treat. Frozen is generally viewed as safer, but you do have to freeze them, which is a bit of a complication.
Tanks- just get standard 5 gallon if you can, but if you see one of those sets, check how much it would cost if you brought everything on it's one. And often, they don't come with heater. And they do have the disadvantage that if a part like the light breaks down, it's hard or impossible to find a replacement without buying a new kit. Just a warning that I have heard, and passed on.
Plants and decor- Check the dangerous ornaments list for ornaments you shouldn't buy, and for you best bet, buy only silk plants. You can also get a cave, buy make sure that your betta can't get stuck.
Jumping- bettas jump. Make sure that you have a lid in your tank, or your fish could jump out and die. Cling wrap is often used temporarily on smaller tanks, with some holes punched in the provide air.
Good luck! And we need pictures of your betta and set-up!