I've seen strawberry pots designed to hang from a balcony or in a window for what amounts to indoor growing. Same with tomatoes. I've also heard that dwarf citrus trees produce well indoors if kept warm enough and given enough sunlight. You might want to consider investing in some good lighting if you intend to grow indoors. It helps a whole lot.
They aren't plants, of course, but I consider mushroom cultivation to be gardening, so I figured I would suggest that as well! Most of the mushrooms which can be cultivated at all (not many can be yet, sadly) can be grown indoors. I strongly suggest oyster mushrooms and shitake as being the easiest for a beginner. They're both very tasty and extremely good for you as well!