A 1 year old should be eating pretty much anything and everything older kids and adult are eating. A wide variety of healthy food is essential. My youngest is 11 months and currently eating whatever the rest of the family is eating. So a typical day is...
nursing for 10-15 minutes when she wakes up
Breakfast is bowl of oatmeal and fresh fruit (bananas, peaches, nectaries are her favorite) or scrambled eggs with toast or french toast or pancakes or cold cereal like Mighty Bites, Cheerios, Alpha-Bits, etc.
Lunch is cheese, turkey and crackers with fruit or leftovers from the night before or grilled cheese sandwich with tomato soup... the options are pretty endless with lunch and dinner.
Dinner is, again, whatever we have.... we have chicken alot (she has no teeth yet so I cook until very tender and shred it for her), a wide variety of veggies (cooked/steamed until soft), any type of pasta with sauce, etc.
Snacks tend to be things like dry cereal, pretzals, gold fish crackers, fresh fruit, steamed veggies or toast.
She nurses before her afternoon nap and again when she wakes up (long 3 hour nap). She also nurses before bedtime (start of bedtime routine) and upon waking in the morning. She drinks water from a sippy cup at meals and has a sippy cup of water nearby throughout the day so she can drink when she gets hot/thirsty.
She's doing what we did with our other kids and they are very healthy and thriving in grade school -- no allergies or anything like that. None of my kids have gotten a tooth before they turned 1. It's not overly common but it's not unusual, either. It certainly has no affect on what they eat since babies don't chew with their front/first teeth anyways... they can gum pretty much anything.