Grand View camp ground is a nice place to base for White Mt. Its about half way up the paved road. If I remember right there is no water there, so tank up in town. On the road out to White Mt there is a great swimming hole on north side, Owens River, where the road crosses the river. Great place to cool off and get the dust off.
A nice place to hike around is the Bristlecone Pine forest your first day. Start your acclimatize program there.