A rabbit's diet should be made up of fresh leafy greens, good quality pellets, unlimited fresh hay (timothy, broe, oat hay, or orchard grass), and water. Anything beyond that is "treat" and should be given in limited quantities. This includes carrots!
Here are a few rabbit-safe vegetables, with more to be found at The House Rabbit Society's page: Kale, cilantro, dandelion greens (these are the most popular!) Dill, basil, mint, turnip greens, red/green/romaine lettuce (no iceberg lettuce), carrot tops, arugula, baby bok choy
Below is information regarding different diets bunnies require at different stages of their life as well as some websites and other resources for ordering hay and pellets.
Babies (Birth-7 months)
Birth-3 weeks: Mother's milk
3-7 weeks: Mother's milk, alfalfa and pellets
7 weeks-7 months: Unlimited pellets, unlimited hay
12 weeks: Introduce vegetables (one at a time in quantities under 0.5 oz
Young Adults (7 months-1 year)
Introduce timothy hay, grass hay, and oat hays; decrease alfalfa
Decrease pellets to 1/2 cup per 6lbs body weight
Increase daily vegetables gradually
Fruit daily ration of no more than 1oz to 2oz (2tbl) per 6lbs of body weight
Mature Adults (1-5 years)
Unlimited timothy, grass hay or oat hay
1/4 to 1/2 cup pellets per 6lbs body weight
Minimum 2 cups chopped vegetables per 6lbs body weight
Fruit daily ration no more than 2oz (2tbl) per 6lbs body weight
Seniors (Over 6 years)
If sufficient weight is maintained, continue adult diet
Frail, older rabbits may need unrestricted pellets to help them keep their weight up
Alfalfa can be given to underweight senior rabbits, but only if calcium levels are normal
Annual blood workups with a rabbit savvy vet are highly recommended for geriatric rabbits