I run through hills. I veer around mountains. I leap over rivers. And crawl through the forests. Step out your door to find me. What am I?

The answer would be a road.

Hello

There are quite a variety of animals that fix your explanation, but the most common one is a squirrel. A squirrel fits your description, except i'm not sure about the jumping over a river. I believe for that, it might also be a stray cat.