I went camping last week up near Mammoth. Actually right on the edge of Mammoth Lakes town at a USFS campground named New Shady Rest Campground. It was pretty nice. The low temps (around 80F) at 7,800 feet elevation were a welcome relief from the 100+ temps down in Pasadena.
This campgorund had an array of conveniences that I am not accustomed to. OK, not the campground exactly. Being on the edge of town, you can walk to shops. It's located across the street from a McDonalds! I bought a sandwich at Subway for a mid day snack while off-road driving one day. It rained lightly one afternoon, and that was refreshing.
The campground itself is pretty nice also. Mildly hilly and the sites aren't too close together. By comparison, the South Campground at Zion NP is flat and most of the sites are right next to one another.
The squirrels and ground squirrels are very habituated to people and they walk right up to you (and onto you) to beg for food. There were no signs of bears when I was there, but the campground host told me that they are often visited by the same bear, mostly checking out the dumpsters.
One afternoon I explored the dirt roads in the forest on the east side of highway 395 (the circled areas on the map). They ranged in difficulty from class 1 to class 3 (Mitchell scale).
The northwestern area indicated has many USFS roads of differing quality. The larger ones were being used by mountain bikers. I encountered many bikers out there, so I had to drive slowly and approach blind corners with caution. I also came upon a gun firing range with many people shooting at the time. That explains the loud sounds I'd heard when I was about a mile away.
I stopped off to gander at the hot springs site. I didn't stay long and didn't walk down to get near the water. They've got signs posted everywhere warning about the hazards and the prohibition on entering the water.
If you're interested in camping in the area, then the USFS has a very helpful guide: 2011-2012 Inyo Forest Visitor Guide (6MB PDF file). I found the link to this on this USFS site for the Inyo National Forest.