FYI, campfires and overnight camping are not allowed in the Bristlecone Pine Forest.
There are 2 main Bristlecone groves. The lower, Schulman, grove has a temporary visitors center. There is a fairly easy loop trail at the lower grove.
The upper grove, the Patriarch, is at the tree line and can be challenging if you are not acclimated to the altitude.
The times I have been there have been in the afternoon and evening for moonrise and sunset. I don't have any specific locations to give you, but in addition to the Bristlecones, there are some spectacular views west towards the Sierra Nevada crest and also to the east for moonrise.
As you probably know, the Sierra had a record snowpack and there are flood warnings for the creeks, etc. You may want to contact Inyo national forest to check on road conditions up to the White Mountains and any other areas.
Paul