When you bring carrots home from the store, if you cut the tops off of them they will last much longer. The stem soaks up all the water from the carrot, and that's why they get limp after a couple of weeks in the fridge. I learned this from the little tip section in a cookbook--really helpful place to find good info!
Celery will keep for weeks and weeks if you wrap it in foil. Don't make the mistake I did though, rinse them and leave a damp paper towel in there with the celery sticks. (found that nice little surprise today)