OK, then let me share some of my favorite north coast places with you... (I've been up there quite a lot!)
There are two "best" areas. The southern one includes Humboldt Redwoods State Park, which is (in my opinion anyway) the best of the redwoods parks. There is plenty of inexpensive lodging in nearby Eureka or Garberville. Eureka is more historic, with some interesting old Victorian houses too. The best part of Humboldt park is the Rockefeller Forest area.
The northern good area has Jedediah Smith Redwoods State Park (Stout Grove there is *almost* as good as Humboldt), Redwoods National Park (which isn't better than the state parks, but about the same, despite being a national park instead), and Prairie Creek Redwoods State Park (where Fern Canyon is the best part). There is plenty of inexpensive lodging in Crescent City, near those parks.
As others have mentioned, California is BIG, so you'll be doing a lot of driving. To go from LA to No. Cal., Interstate 5 through the central valley is very fast, but very, very boring. Highway 101 (nearer the coast) is more interesting, but much, much slower. The mapquest web site is good for finding distances and driving times. Highway 101 is what you want to get from San Fran to the north coast; it's not fast, but it's very scenic.
If you're short on time, you can do just the southern area (around Humboldt park), skipping the northern area, and not miss too much.
Best of luck with your trip planning, and let me know if you have any more questions.
Lisa