The betta can breathe some air, so it can manage with a smaller volume of water than some other fish can. It would be nice to plan on a larger tank at some point, because they like to swim when they have room to do it.
As far as water changes, when you first set up your tank, it has to cycle and may need more frequent changes. You can get some bacteria to culture your tank at the pet store (like this
Top Fin® Bacteria Supplement Aquarium Cycling Water Conditioner | Cycling Aids/Bacteria | PetSmart ) , or get some gunk from a friend's tank to get the right bacteria growing in there.
I would change a gallon every week at first, more often if it's cloudy or really smelly. The danger is that the ammonia levels in the tank will get high enough to be dangerous to your fish before the good bacteria grow to start eating the ammonia. The smaller the tank, the more problems you can have with water quality.
Once you notice it having less odor, you can gradually space the water changes out. You don't ever want to change all of the water at one time, and even changing too much water too often can mess up the balance in your tank.
When you do get a bigger tank, put some of the water from the old tank into the new tank along with your fake plants and that will help the tank cycle sooner. A ten gallon tank won't take up much more room than the three gallon, but he'll have room to really swim around.
Please register to participate in our discussions with 2 million other members - it's free and quick! Some forums can only be seen by registered members. After you create your account, you'll be able to customize options and access all our 15,000 new posts/day with fewer ads.
Also...not sure if makes a difference or not but I have a couple fake plants, one real plant and one suction cup leaf that they like to lounge on in there as well. 
