Start off with keeping them as pets before you think about breeding. You need a considerable amount of knowledge about betta genetics and quite a bit of money and time to spare if you want to breed successfully.
A basic betta set up would consist of:
5 gallon tank - longer than tall and avoid novelty tanks like Fluval Edge and BiOrbs for first time. Also avoid tall, thin tanks. Smaller tanks are do-able but I'm a big believer in giving all pets - whether stick insects, fish or horses - plenty of space for a stable environment, exercise and stimulation.
Cycled filter - this is optional but without one you need to be doing 100% water changes 3-4 times a week which is more stressful for the fish and more work for you than a filtered, cycled tank. Your filter should be gentle and without a strong outlet flow.
Heater - bettas need temps of 26-28C.
Very soft net or cup - regular nets can easily damage their fins
Plants - live or silk, but they need them in order to feel secure and have a natural environment
Smooth gravel or sand - they like to sleep/sit on the bottom of the tank and some gravel can be too rough
Species-appropriate food - I like Atison's Betta food for a staple. Some use Hikari (I used to) but there are some concerns about their new recipe. They also need regular feedings of live and/or frozen food such as bloodworms and brineshrip. These give them the dietry variation they need and help stop them getting conditions associated with too much dry food (such as swimbladder disorders).
A cave - smooth, hollowed out coconut, terracotta pot . . . whatever really. Just make sure it is smooth and doesn't have any small holes they could get stuck in!
First-aid kit - aquarium salt, some basic meds for bacteria, fungal and parasitic infections, spare filter pads, carbon pads and other emergency supplies.
Test kit - essential for all fishkeeping is a liquid test kit for ammonia, nitrite, nirate and pH. Hardness ests (KH/GH) are also preferable.
Gravel vacuum - again another fishkeeping staple. Essential for keeping the gravel free of poo and uneaten food that can decay and poison the tank.
Buckets - that's just common sense, really.
The right water - again as with all fishkeeping, you never want to assume your tap water is suitable. Bettas are fairly tolerant but do much better in low pH, soft water. I have a lot of trouble with mine in a very hard water area - I think it makes them less hardy and more prone to diseases.
This is my suggested set-up for a single betta splendens, non-giant variety. If you want some wilds (such as imbellis), you'll need to adapt your tank-size accordingly.