I'm going to talk some basics for a while, if anything confuses you let me/the other posters know :) Also, if this starts getting boring and covers things you've already researched, awesome! and just skip the silly thing.
We've talked a lot about cycling tanks. A cycled tank is a tank that has bacteria living in the substrate(gravel or sand at the bottom of the tank) and in the spongy mesh that should be a part of your filter cartridge. Very little of this bacteria lives in the water floating, it remains attached to the rough hard surfaces that I listed. These bacteria are a specific type of bacteria that effectively "eat" two compounds called ammonia and nitrites. Ammonia comes from fish poop, as well as any dead "organic" matter in the tank. this can be decomposing food, dead plants, or other dead fish. This ammonia is turned into nitrite.
is then turned into nitrate
by a different bacteria. Nitrate
is less toxic to fish, and is then removed when you do a water change, or is used as food for any live plants you may have. In a newly set up tank like yours, these bacteria have not started to live in the gravel and filter, or there aren't enough of them to keep up with the amount of waste your fish produces. This is why it is important to change out some of the water fairly frequently, because the poisonous ammonia and nitrites are not being changed to the less poisonous nitrates. The test kit can tell you how far along your tank has cycled. First, the bacteria that "eat" the ammonia will start to grow. This will mean that when you test the water, the ammonia levels will be low. Around this time, the nitrites will start to increase, because only the first type of bacteria has established itself, and so only the first conversion is happening. The next thing that will happen is that the nitrite levels will drop, and the nitrate levels will rise. This is good. This means that now you have both types of good bacteria working to keep the tank healthy.
Like the other users have said, the only way to get a tank cycled is, unfortunately, to wait. In the mean time 50% water changes every other day are whats suggested for your size tank in the betta care FAQ sticky.
Just thought I'd talk about the cycle a bit, I dunno how much you've read up but I figgered I'd put it out there because it can be quite confusing.
About changing the filter cartridges in your filter: I have filter cartridges that come with "activated carbon". This is a type of charcoal that absorbs and keeps compounds out of the water. Different people say different things, but I have heard that keeping a cartridge in too long may lead to the carbon leaking these, potentially harmful, chemicals back into the water. Some people suggest that carbon is only necessary when removing medicines or in the presence of an odor problem (though this is usually caused by a problem that should be addressed). If you want, you can remove the charcoal from the filter if it has it by cutting the back and simply picking it out. Then you can keep the filter cartridge and all the good bacteria that live in it without risking endangering your fish. Personally, I remove the cartridge every month since my activated carbon is hard to remove and my filter has whats called a "bio mesh" in it. This provides a home for the bacteria, and when new looks like like a black brillo pad. If you don't have one of these, which I'm guessing is the case based on the size of your tank and the filters that normally work for that size, I would suggest removing the carbon/charcoal if possible to avoid the risk of harmful chemical leaching. Then you can just rinse the remaining spongy mesh in a bucket of tank water and pop it back into the filter.
Right now you are doing good, keep testing the water for the compounds I mentioned, and keep up with water changes.
Also, also, if in the future you have the space, resources, and the desire, I highly recommend upgrading to a 5 or 10gal tank, and this time around you'll be ahead of the game and can cycle before moving your fish. I have a 10 gal and its great, the betta I have in it really came alive once I put her in the tank and she had room to explore.
And here's a link to the betta care sticky if you haven't found it already: http://www.bettafish.com/showthread.php?t=20058
Sorry for that novel, I just figured that more information couldn't hurt. I hope things turn out well! But be forewarned, once you've been bitten by the betta bug, you're DOOMED to a life full of the pleasures of keeping these wonderful fish