You would be surprised at what they can already eat!
My DD (13 months) eats pretty much everything we eat. She loves veggies - green beans, carrots, corn, potatoes, sweet potatoes, bell peppers, peas, asparagus, etc. Fruits we do strawberries, bananas, pineapple, small pieces of apple, you could do melons (we're staying away from them for a while bc of family allergies), blueberries, etc.
Other things she likes...breads, cheeses, macaronni and cheese, pork ribs, meatballs, spaghetti, ravioli, deli meat (bologna, ham, turkey), chicken breast, pizza.
Like I said, we feed her just about everything we eat. She only has 8 teeth and has been eating table foods for about 5 months now. You didn't mention how old your kids are but if they are wanting to feed themselves and one has 16 teeth then I would think they are old enough to eat most of what our DD eats.