I'm sorry I haven't read all the replies but hopefully something in this may be useful to you:
5-10 gallon tank for a single betta fish is fine. If you want multiple female bettas then a 20 gallon or larger. For a small tank 10g or less you can use an air pump and sponge filter. You can buy valves for the air line to control the flow.
Get your aquarium, heater, thermometer and filter setup before you buy a fish and I highly recommend an aquarium water test kit (or take water samples to you local fish store to test but that's so tedious). Setup the aquarium, fill it with conditioned tap water, start the filter. Wait 24 hours. Add ammonia and TSS+ beneficial bacteria to the water. This will get your cycle started. When the cycle processes 4mm ammonia in 24hrs you are ready for fish. The cycle takes 3-4 weeks or longer to finish. For a betta fish you don't have to wait until the cycle finishes to add the fish because these are usually very hardy fish. But do not let your ammonia plus nitrites get over 1ppm total or the nitrates to get over 20 (I think it's 20, it's been awhile), do 25-50% water changes before the numbers get to that point.
Once you have your first tank cycled you can throw in extra filter material, then if you ever need to set up a second aquarium (or hospital/quarantine tank) you can just use the extra filter material for the new filter without having to cycle the 2nd tank. Also keep in mind that betta fish like warmer water about 80 degrees and warmer water speeds up the cycling process too. I did dirted and planted tanks, the dirt releases ammonia for the first little while so that combined with seeded filter material is how I started and cycled all my tanks.
When you do your weekly water changes condition the tap water with seachem prime (liquid) or seachem safe (powder - lasts forever). When you shop for a fish, take a flashlight with you or use the flashlight on your phone. If the scales look like they have flaky gold flecks on them, do not buy this fish. Do not buy a fish that has visible white spots or if it looks unusually thin or if it has a kinked/bent back. All signs of disease that will be difficult to treat and could be fatal.
So your shopping list would be:
5+ gallon aquarium for 1 betta
heater
filter + filter material (just rinse with old tank water and reuse the filter material)
gravel/decor/plants (only silk or live plants for a betta fish with long fins)
thermometers
aquarium liquid test kit
TSS+
Seachem Prime (seachem also makes bottled beneficial bacteria that works better with the prime I forget what it's called though)
Prazipro is a good parasite treatment to have on hand.
If you really want to blow your mind check out Dustin's Fish tanks on youtube on how to set up a dirted planted tank. Dirted and planted tanks... once you have the tank cycled... are the most stable and forgiving set ups IMO and your fish will love them. Also I'd like to play devil's advocate and say look into Nano fish such as pygmy rasbora as an alternative to a betta fish if you want more than one fish in a small tank. HTH