A sponge filter is basically what it sounds like, a sponge with an air lift. The sponge is attached to a tube and an air pump, the bubbling water sucks water through the sponge. These are good because they cause very little current, and for low bioload tanks they're very effective. The drawback is they tend to be rather big and unsightly, some people also don't like the noise of the air pump.
Power filters pump water up the intake tube, so that the water spills over into a reservoir that holds filter floss and carbon, this is where your bacteria lives. The filter's job is to circulate water over the floss so the bacteria can process the fish waste in the water column. These filters are less obtrusive than the sponges, but they can create a lot of current and some people don't like the waterfall noises they make. If you don't mind the noise, you can easily create a baffle for your filter out of a soda bottle. Personally, I never change the cartridges in my power filters--this is where your bacteria lives, you don't want to throw that out. The reason the company wants you to replace them is because the carbon is ineffective after about a month--your aquarium doesn't really need carbon, so you can just leave the cartridge in there, rinsing it off every once in awhile. I like using fresh carbon, so I usually cut a slit in the floss, empty out the old carbon, and replace it with new higher quality carbon.
Canister filters are similar to power filters, except the reservoir they use is a closed canister with a lot more space to put in filter media, some are large enough to put heaters in. These often have a spray bar rather than a waterfall outlet like a power filter has, this makes them extremely quiet. The plus with these is that they are very customizable, you can point the current anywhere, and fill it with whatever you want. The downside is that they don't aerate the water, so when you cycle with a canister you must add additional aeration.
I suggest doing your research on different types and looking up reviews on different brands before you go shopping. This site is a great resource, lots of variety and very good prices: http://www.fosterandsmithaquatics.com
even if you don't want to buy online, this is a good way to look up possible products that you can find product reviews for elsewhere.
As far as cycling, this isn't something you really can just let happen on its own. You need a liquid master test kit, this is not optional no matter what method you use to cycle. I prefer doing the fishless method, because you get a more stable bacteria colony, it's less work, and it's the kindest to your fish. You would have to put your fish into a temporary container, such as a rubbermaid/sterilite plastic storage bin (4 gallon bins are like $3 at Walmart or Target) and do regular water changes on that container while the tank cycles. These containers are ideal for bettas and can be safely heated. They also make great hospital tanks since you shouldn't use medication in a cycled tank, and they're also great for quarantine if you decide to get new tank mates at a later time. This sounds kind of like a pain in the butt, but remember, if you do it right, you only have to do it once. It's worth it.
You should research the nitrogen cycle on your own so that you fully understand what goes into cultivating and maintaining a bacteria colony. I really don't want to type it all out right now, lol.