Alabama Hills is a great place to explore. Also a good choice for a first primitive cap. Its located just outside of town.
Just follow the movie road around and find a nice place for your basecamp. Try to find a spot among the rocks for shelter as the wind can be brutal at times, especially from the west.
There are roads and trails and mines to explore all around you.
The trip to Cerro Gordo is cool too. If the road is open up to the Whitney Portal be sure to check it out. Highly recommend the breakfast or hamburger at the Portal store. Have a great time.
Chris