Here are some hikes for early in the season (starting June or even May in some areas, this low-snow year):
Maybe focus on these for now while you acclimate to the altitude? Most of these destinations are below 8,000 feet.
They should all be within your 4-hour drive limit. Drive in very early morning for best results. On a summer weekend, any departure from home after 9-10 a.m. can add an extra hour to your driving time. Friday nights, it can be all tied up.
I think the biggest bang for the buck in terms of short-drive-to-high-mountain-scenery ratio (from Santa Rosa) is the Desolation Wilderness. As few as 3 hours door-to-door depending on trailhead and time of day. You WILL however see a whole lot of other people there! And, you will have to indicate on your permit which narrow "zone" you intend to camp at each night. Elevation ranges from about 6500 to 10,000 feet depending on where in Desolation you want to go.
Some of the driving times to places are counter-intuitive. I have gotten to Lodgepole trailhead in the SEKI in just 4:30 from Oakland. Some of the trailheads in Emigrant Wilderness, though far closer (in miles) to the Bay Area, are the same 4:30 drive, because you end up on windy and/or dirt roads.