First, I want to point out that if a person were to go too long without mowing, there can be
enough grass so that when one mows, it could clump or mat and that would be bad for the lawn. If you mow often enough, this won't happen and the clippings will fall between the grass blades and
feed the grass.
Leaves: when you have a few leaves, mow em! The pulverized leaves will also fall between the blades of grass and then feed the grass and the tree! This is one of the big reasons why trees play such a huge role in
permaculture! The annual leaf drop is one of the best ways to feed the soil!
If you have a huge leaf drop and you have a mulching mower, you might be able to mulch the leaves enough to fit between the grass blades, but probably not. In this case, It would probably be best to compost your leaves. OR! if you have a garden, or some areas where you do not want grass to grow, you can use the leaves as mulch!
As for twigs and the like: for anything big, I pick it up. The little stuff will decompose fairly quickly.