Cutting up the whole thing will save you time and dishes in the future, and on tired nights, not having to do as much meal prep can mean the difference between a home cooked meal and a dinner out.
Once they are cut, onions do just fine for a week or so in the fridge. Carrots and celery, on the other hand, need a little help if they are to last after they have been cut. I like to add a layer of water in the bottom of the container for vegetables like carrots, because it helps them stay hydrated. A quick shake of the container and all the carrots will be moist and ready to eat or use in recipes.
You can cut most vegetables ahead of time, just as long as you make sure they have a good airtight container in the fridge. We like to have broccoli, cauliflower and cucumbers cut up and ready in the fridge, because we can snack on them throughout the day or toss them into a stir fry.
Though it takes more time to cut the whole vegetable, you save a lot of time later by not having to handle the knife and cutting board, and you have the convenience of just taking them out and using them right away.