A one-year old should be eating whatever the rest of the family eats. And he should be 100% self-feeding with his fingers with a little help for a spoon. My older 2 kids were 100% self-feeding what the rest of the family was eating by 10-11 months. My 9 month old is almost there, too. My older 2 kids didn't have a single tooth until after their 1st birthday and the 9 month old is currently toothless, too. There's really nothing a 1-year old shouldn't eat except maybe peanuts because of the choking hazzard. If he doesn't have front teeth yet, which are used for biting things off, then just cut everything into pea-size bits. Babies can gum pretty much anything. Most babies don't get the back teeth for chewing until well into their 2nd year anyways.
So anyways... A typical day would be dry cereal with a sippy cup of milk and fruit slices/chunks (banana, pear, peach, grapes cut in half, melon, apple, etc). Lunch can be thin-sliced deli meat like turkey or ham finger-shredded into small bits, fruit, steamed carrots and shredded cheese. At about 1.5, my kids were eating real sandwiches cut into small pieces. Dinner, again, is whatever the family is eating -- pasta, chicken, casseroles, salmon, any veggies, any fruits, rolls, biscuits, toasts, pizza (we makes our at home so it's healthy), etc.
My toothless 9 month old eats all of the above. I just make sure the meat is slow-cooked until it's fork-tender and easy to fork-shred. I don't believe in making special "kid meals" becuase then the kids get into the habit of being 'waited on' and, to quote my mom "I'm not a restaurant!". :-) Becuase the rest of the family eats healthy, there's no reason my baby can't eat that too. I guess it would be different if we ate junk, but we don't.