My son will be one on Monday and is quite into feeding himself so I was interested in this too. Here is some of what I've tried as well as some of what I found on the internet. Anything I mention should obviously be cut into chunks or slices.
avocado, bread, any kind of beans, pasta, graham crackers, peeled and sliced grapes, scrambled egg, tofu, cooked brocoli, cauliflower, edamames, peas, corn, waffles, pancakes, french toast, pieces of burger, cucumber, tomato, peeled apple, dried fruit like raisins, apricots, apple, banana, pineapple, mango, and papaya, squash, yam, hot dog, olives
If you like to have quick things on hand, check out the frozen food section and canned goods. Lots of quick veggie and fruit ideas.
Pretty much if it isn't hard like a raw carrot, you can serve them what you are eating, just in small chunks.