You can get to the bristlecone forest no problem -- road is paved. Past that, the road is not paved if you want to get to the Barcroft gate, which is the trailhead for White Mountain.
If you go early in the season, the road to Barcroft may not be possible in a regular vehicle. However, they have an open house the first Sunday in August, and they try to get the road ready for that occasion. Last year, a friend and I did the trip on August 1 in a Volvo stationwagon -- high clearance would have been nice, but it was totally doable, and we saw some sedans doing it to.
One benefit of going the day of the open house is that they open the Barcroft gate so you can drive up to the lab. That means that the roundtrip hike to the summit of White Mountain is only 10 miles, rather than 14. But even if you go after the open house, you should benefit from the road work they do to prepare for it.
You can camp at Grandview if your goal is the bristlecones. Alternatively, you can car camp at the Barcroft gate if your plan is to hike up White Mountain. There is no water, but there was a pit toilet there last year at the time of the open house at least.
