Well...
Well, the minimum tank/bowl size that is acceptable is 1gal, but a more ideal size is 5 or 10 gals. Depending on where you get your tank, this could cost anywhere between $10 and $30 or more (apparently Petco, in the United States, frequently has a $1/gal sale on their tanks, or you can purchase a kit that includes a hood, filter, etc for a little more).
You're going to need a nice quality adjustable heater, which will run you anywhere between $20 and $40, depending on the modle, however, I suggest not skimping and buying a cheap-o heater as this will be the product that is most likely to cause a fishy fatality if is malfunctions. My tank came with a heater,so I can't really offer suggestions as per the best model, but I'm sure another member will pipe up for you! Also, I suggest checking Amazon.com or Ebay for heaters as they are cheap to ship and are often *much* cheaper on the internet than in stores. You will also need a thermometer. I suggest a glass thermometer that suction cups to the inside of your tank, they are about $3.
You can get a filter if you want, but that is optional. there are many different types that filter with different methods. A word of caution about filters: if the current that your filter puts out is too strong, you will have to baffle it, which reduces the current. A step-by-step filter baffling instructions can be found in this thread
http://www.bettafish.com/showthread.php?t=30139 . A filter will also allow you to cycle your tank. More information on cycling can be found in this thread:
http://www.bettafish.com/showthread.php?t=107771 . If you plan on cycling your tank (which makes it much easier to maintain water quality) You will need to set up the tank and begin cycling about a month before you plan on adding the fish.
You will need some sort of substrate and decor. Personally, I have an NPT, or Naturally Planted Tank which means that I have a layer of organic potting soil covered in a layer of substrate (gravel in my case but many people prefer sand) with tonnes of live plants. There are many benefits to this system and it looks great, is fairly easy to maintain and I sell my plant clippings on craigslist, so I've already made back what I spent on plants by selling my overgrowth! This system seems really daunting to set up, but once is gets going, it's really simple. Alternatively, many members like to use silk plants as hiding places. You'll also want some sort of cave or hidey-hole. For this, if you're trying to save money, you can use a simple terracotta pot or mug (that has never been used, or had soap used on it) turned on its side and slightly submerged in your substrate. Or, you could do what I did and make a cave using boiled (to sterilize them) rocks and aquarium grade silicon. There are also MANY other decoration type available online and in store that are designed specifically for aquariums.
A few tips when purchasing decor for your tank:
1) no metal: metal will rust in the water, polluting your ecosystem and potentially harming your betta
2) Check for cracks, chips or flakes in paint: this is a good indicator of workmanship and quality.
3) No rough edges: bettas have very delicate fins that can be damaged or torn very easily. A generally safe test if the pantyhose test: you drag a pair of ladies pantyhose over every conceivable angle of the decoration and if they snag, it will likely damage the betta's fins.
All the decor can cost anywhere from a few dollars to hundreds, depending on what you want.
You will also want a good water conditioner. I suggest Sechem Prime as you need only 0.1ml/gallon of water and it's about $10 for a 100ml bottle that will last you a *very* long time. Other members also really like the API Stress Coat as well. This will remove any chlorine or chloramine from your tap water, making it safe to use in your aquarium.
The final thing that you will need is a water testing kit. A liquid test kit is the most accurate and over time, is the cheapest as they last longer. It will cost you $30-$40 up front, but costs less than $0.50 per test and will provide you with valuable information about your water parameters. The gold standard on this forum is the API Master Test Kit (freshwater). It provides you with tests for nitrate, nitrite, ammonia, pH, and more.
Realistically, you are looking at about $100 to get your tank fully set up and running. But it could be more or it could be less depending on how hard you look and how much you want to spend
I hope I was able to provide you with some useful information and answer your questions
Edit: you'll also want a good quality pellet food, like Omega One Betta pellets or New Life Spectrum Betta formula, about $10. And a siphon or gravel vacuum, about $5.