Passive writing has its uses (I’ll try to get around to a blog post on that subject at some point), but, in general, portraying activity and motion engages the reader better. So, some tips:
1. For the love of all that is considered Good Writing, refrain from sentences like this, “Joe was climbing the mountain.” Translate that as, “Joe existed in a state of climbing the mountain. That’s horrible. Readers want to see Joe climb the mountain, not Joe existing. “Joe climbed the mountain” is more active and uses one less word.
2. Take a portion of your work — be it a chapter, a scene, or even a paragraph — and write down just the verbs. If you have a list that mostly conveys activity and motion without a lot of repetition, you’re doing great. If you see a bunch of was, could, had, see, look, and heard, then not so much.
3. If you see that your character started to run or it began raining, consider if you need “start” and “began.” Sometimes those words are, indeed, necessary. In a lot of instances, though, they’re a wasted opportunity. Consider: It started to rain. Would “It rained” convey what you need? Better yet, how about: A raindrop, cold and wet, splattered on Joe’s head. He cursed. The preceding both conveys that it is starting to rain AND, more importantly, adds the filter of emotional context to the event.
4. Inanimate objects can be described actively as well. Tolkien’s trees “marched” down the hillside. Mountains can stretch to the sky. Trees can loom over you. Even having a castle stand, though it doesn’t convey motion, is slightly better than having the castle simply exist (was).