Hello and welcome the forum! And it's great that you're doing your research ahead of time rather than playing catch up.
A 2.5 gallon is great. If you want a good deal on a tank and filter, though, I'd recommend getting the $30 5.5 gallon kit
from Walmart. It will likely be much cheaper than the equivalent from Petco (and probably almost the same price as a filter alone).
If you do go the 2.5 gallon route and do 100% water changes, I would take the fish out and put him in the cup that you buy him in. Make sure you put the cover back on because they DO LOVE to jump!!!!
No, don't clean with anything special. Soap is toxic to fish. The only thing you need to clean with is hot water. And just rinse everything (tank, gravel, decorations, just not the fish :-p)
As for a heater, you really should invest in one. Even though your temp will be pretty warm in your room, the water always stays cooler than room temp. And your temps will fluctuate at night, which will make the water temp fluctuate. Bettas regulate their body temps by the water temps, so going up and down will make their body temps go up and down, lower their metabolism, lower their immune system and make them more susceptible to illness. Also, having a constant temp of at least 76* (and it should not be any lower since they are tropical fish) will make your betta more active and thus more fun to watch. Here is an inexpensive heater
that I highly recommend (it is adjustable and I have purchased from this seller in the past and had great service and quality).
You will need to add conditioner each time you do a water change. I use, and recommend, Prime which is top quality. It seems expensive compared to the others, but you only need to use 2 drops per gallon so it ends up being much cheaper in the long run.
Feel free to ask any other questions you may have!