He is driving to the mountains.
This means he is driving toward fhe mountains and probably will get there. It does not specify whether he will stop at the base of the mountains. If you wanted to make it clear that he will carry on when he reaches the base of the mountains, you would say
He is driving into the mountains.
This would probably mean "entering the mountainous area", not driving inside the mountains for example into a tunnel.
For your sentences, it is not necessary to use reach. You might, however, say something like this, meaning that he won't stop until he arrives:
He is driving until he reaches the mountains.
It is not necessary to use to with the verb reach when you mean arrive.