Get yourself a brand name hang on the back filter. There are good ones from Hagen, Marineland, Penguin, ATI. There are many and they are all fairly cheap, just get a good one and you wont replace it. Make sure its rated for between 50-100% MORE than your actual water volume. The filter will consist of a filter pad and ceramic media or sponge. The ceramic / sponge is where the bacteria will live. The pad is the part that traps debris and gets replaced regularly. If you can get somebody's old filter pad it would help in kick starting the bacteria in your tank. If not, they will appear eventually any way.
Until you have a working, thriving bacterial colony, you should do lots of water changes. 25% every other day would be good to prevent the buildup of poisonous ammonia. Test your water daily for ammonia, and reduce the water changes once the ammonia spike disappears.
By now you should should be familiar with cycling a tank. If not find those posts and read them. The bettas are very hardy fish and will be fine, hopefully the corys will be as well. Just go slowly.