Did you particularly want to DRIVE thru old growth redwood forests, or did you just want to see groves of redwood trees?
Muir Woods is the closest old growth redwood grove (i.e. TALL trees) to San Francisco, and is across the Golden Gate Bridge on Highway 1 (about 30-40 minutes at most). However, you must park your car at the gate and proceed on foot to see the trees (not problematic, since the trees start at the entrance gate). Parking is a major problem at this location however, as the parking lot is tiny as compared to the throngs who visit -- consequently you may find yourself parking alongside the road and walking to the entrance.
Armstrong Redwoods State Park is close to the Russian River in Guerneville (pronounced GERN-ville), and is significantly less touristed than Muir Woods. This approach is about 1.5 hours north of San Francisco (take Highway 101 to Santa Rosa, and take the Guerneville/River Road exit). You are able to drive thru this smallish park on a short paved road, although there are no trees you can drive thru. This park does have numerous hiking trails though. The nice thing about the Santa Rosa/Guerneville area, is that you can take in some wineries in the area (Sonoma County). The Korbel champagne (oops! sparkling wine) cellar is on this route. Plus you can follow the road from Guerneville to Jenner (about 20 minutes, where the Russian River empties into the Pacific Ocean), drive a few short scenic miles to the south, and Bodega Bay is before you (if on a fog free day, that is). This is where Alfred Hitchcock filmed "The Birds." Take Highway 12 back to Santa Rosa/Highway 101, and you've taken a nice, compact, scenic route representing the best of Northern California.
Alas, I've never made it up to the Avenue of the Giants. But you're going to experience the Redwood Empire in all its glory up here. North of Willits, at the point where Highways 101 and 1 meet up again, there is a tourist trap place, where the owners proudly proclaim you can get a picture of your car driving thru a redwood tree, but I didn't pay the $10 or so it cost for the "privilege." As far as driving time, I have no idea, but I drove from Santa Rosa to Hopland on Highway 101 last week, and it took about an hour. When you look at a map, that segment may give you an idea of how long the journey may take in total. Plus Highway 101 is not a freeway for the entire distance, so you're passing thru several smaller towns with traffic lights. This will certainly add to your time spent driving.
Sorry so long-winded, but I enjoy driving thru this area, and I'm glad to provide any help you may need.