Hey Frogi. Well here are a few personal answers from others.
"It's always recommended that you do a fishless cycle, but with a tank like you're going to get, no, it's not completely necessary. You can cycle the tank with the betta. Just be sure to get yourself a good freshwater testing kit and keep an eye on your ammonia and nitrite levels so you know when to do water changes before they get to toxic levels. People will probably vote me down for it, but it can be done safely."
"While it's better to cycle the tank, a betta will be OK just dumped in and letting the tank cycle with the fish in there. They are quite hardy (tolerant of a little ammonia) and don't produce a lot of waste. Do a few extra water changes for the few weeks just to be sure.
Cycling IS important, but if the tank is large (compared to the amount of fish), and the fish are hardy, then it's OK to cycle the tank with the fish.
Live plants also help as they will absorb some of the ammonia the fish produces."
"I have seven bettas, all in different types of tanks... Recently I got one without getting a tank first so I had to put her into an uncycled tank. I actually had fewer problems than with the bettas I put into a fully cycled tank... (As soon as they went in, the nitrite/ammonia levels went up.) Never any problems with the uncycled tank though! She had three plants and even a snail and they're still fine.
I used the product Cycle (which adds the essential bacteria in there), though so that might have helped a lot... That, and the plants. :)"
While not necessary, many people prefer to do it so there is less work involved, just a weekly water change to keep the nitrates down.
When you don't cycle, you have to do frequent water changes to get rid of ammonia build up, which is deadly to fish even in smallish quanities. When you cycle, you are growing bacteria that convert that ammonia to nitrItes. NitrItes are still as deadly to fish as ammonia. The bacteria then again converts the nitrItes to nitrAtes, which is less leathal to fish, and build up slower.
The bacteria that does this will grow on all surfaces of the tank, but mostly in your filter, so you don't want to scrub the tank down, or change your filter, otherwise you're getting rid of the benificial bacteria. To clean your filter, when it gets super dirty, swish it gently around old tank water. Don't worry if it's colored brownish, that'd be the bacteria. :)
It is advised that you do a fishless cycle. Basically, you have a source of ammonia. This can be fish food, pure ammonia, or some piece of raw fish/shrimp (in pantyhose)
After you get it, if you use pure ammonia or fish food, you dose the tank daily. For the raw shrimp/fish you stick it in, and let it sit.
Once a week, you'll test the water's ph, ammonia, nitrItes, and nitrAtes. The best water tester kit would be the API Master Test Kit. (you can order this from Walmart, and have it shipped to a store near you or to your house for half of what Petsmart or Petco sells it for.)
As time passes (I advise you make a little chart or something to track it's progress), you'll see that first the ammonia spikes high, alarmingly so, then drop to zero.
As the ammonia drops, the nitrItes will begin to grow. they will then begin to fall, and become zero.
Then the nitrAtes will begin to grow. When they get high, maybe around 40-80, you have to do a massive (50-80%) water change. Then, your tank is cycled, and ready for your fish to be introduced.
Once a week, you'll have to do a water change (the ammount depends on how stocked your tank is and how large the tank is), and you will have to gravel vac as well.
If you get a API master test kit, and the ammonia is 0, the nitrItes are 0, and the nitrates are present (preferably 5-10), then yes, your tank is cycled. You can't just assume it is because there is brown stuff on your filter.
Oh, I know this is random, but a tip: don't bother buying testing strips, those things are notoriously inaccurate.