Can simple bacterial culture be done at home using food agar as the medium (with some additives found in the store)?
Yes this is possible. There are many sites that show you how to do so, but here is the easiest I found with a brief search
Be careful when doing it that you don't expose yourself to anything toxic though. Seal the dishes and sterilize them before and after. Try to work with cultures that you know and avoid fungal spores