I've been on top of a mountain in a lightning storm, and I don't recommend it. We elected to stay down lower. There are many good hiking trails within the park. I took a group up the trail through the bristlecone forest and to the base of the glacier. A moderately difficult maintained trail stretches for about two miles ending just below the glacier.
Many people push beyond that, right up to the snow.
It's a great hike. There are many back country hiking and camping opportunities within the park, and there are about 30 small caves which can be explored by those who dare. Stop at the visitor center for maps. And be sure to tell a ranger where you are going. It's a great place to get away from the crowds.