In a 5 gallon I would recommend using the filter or aerator anyways. Then when you do water changes do 60% and 80% each week with one of those as a substrate vacuuming and the tank will pretty much cycle itself.
Throw in some live plants, I love Java Ferns and Anubias since they take almost no care at all to stay alive. My Betta seems to love them too so that's always a bonus. They help pull ammonia out of the water and add oxygen back in.
Maybe add in some Ghost shrimp or a snail to clean up after the Betta and any algae that may try to grow and, voila, ecosystem in a box!
Cycling and planting a tank may seem like work but really it isn't. In the long run they actually save you a lot of work.
As for the food, you have to ween a Betta off bloodworms and get them into pellets. The other option is to find a pet store near you that offers frozen food or try raising your own live food.