I'm a fan of prepping some vegetables for the week on a day off. I'm much more likely to eat salad if I have the fixings ready in my fridge. I'm no expert, this is just what I've found works best for me.
Certainly some vegetables are more suited to cutting and storing than others. I find controlling humidity and condensation in storage to be the most important factor. I store almost everything in covered pyrex, unless it needs to breath (and those go in the humid compartment). Veggies with less water can be kept humid with a lightly moist paper towel on top. Wetter veggies can cause condensation and lead to sliminess. A dry paper towel and/or an open container can help with that. A veggie should either be submerged in water, or not in water at all. Sitting in condensation is that biggest problem in storage.
Cut carrots keep in water for a week with no change in quality. Out of water, they can dry out a bit. I don't find the dryness to be a problem.
Sliced bell peppers keep pretty well with a moist paper towel in the container.
I find onions to be pretty indestructible, though the flavor gets a bit milder over time.
Broccoli and cauliflower are also pretty hardy. A dry paper towel or two will deal with water from washing them, which is the biggest problem I've had.
Mushrooms and leafy greens need to breath.
Cucumbers and tomatoes don't store as well as others.