As Marg said, all the comments are good ones, and Frederick's advise about avoiding LA rush-hour traffic in that area could be a life-saver! Try the 210 around LA, or even the highway throught Tehachapi into Bakersfield.
We have traveled Hwy 1 (Pacific Coast Hwy) pulling our Scamp
13 with a Mazda MPV - no sweat. Our recommendation is to drive south-to-north (as you are planning to do). That way, you are on the INLAND side of the roadway, away from the edge where it can 'appear' to be very hairy driving! As you might guess, I am a tad paranoid about mountain-side roads. One other proviso - this road is a magnet for bicyclists, and they sometimes seem to have a 'death wish' - don't seem to care about vehicle traffic, they just GO - arrgghhh!
As others have said, be sure to take advantage of the many pull-outs so the driver can also enjoy the scenery ... when driving, you GOTTA keep your eyes on the road. I'm sooo glad we did that drive, but once was enough, at least towing. We also second the idea of visiting the Avenue of the Giants - it is only 16 miles long. but it took us nearly a week of camping and exploring to get thru it - awesome!
We hope you will enjoy this amazing and gorgeous area of California. We wish you safe journeys and many, many happy memories. Hope you will post so we can all follow your adventures. Take care, L 'n D
Edit - after consideration, only think about Tehachapi if you are coming from I-40 ... that pull over Cajon Pass can be ugly!