Well, at least you're here now and want to know more about your fish! So, welcome!
I'm guessing you bought something like this
? This tends to be the most popular "betta" kit.
Unfortunately, many pet store employees don't seem to know much about bettas. No animal can really "take care of itself" of course, although bettas are a bit lower maintenance than other fish I suppose...
Well, firstly, if you've bought that kit or something similar, those are generally a little too cramped for bettas. Something 2 gallons or above is what's generally recommended. So you're on the right track with that!
While bettas don't require filters, they can still be used. You can read up on the nitrogen cycle here on the forum, there are many useful topics on it. It's a little hard to cycle smaller tanks like 1 or 2 gallons, though. If you don't choose to use a filter, you will obviously have to change the water more frequently, especially in smaller tanks.
Bettas definitely do need a heater! Unless you live somewhere which has a constant warm climate, I suppose. Bettas like their water to be around 76-80 degrees. Your betta's lethargy could be because he's cold.
Lastly...it's pretty normal for bettas to be a little freaked out when they transition! So don't worry too much about that, unless he looks physically ill.