Part of backpacking is learning to work within the permit system. I can relate to your frustration with permits and such on the coast or Bay Area. It seems like this highly populated area does all it can to discourage backpacking with most areas booked weeks in advanced. It is not that bad if you get out of the coastal areas.
There are reserved permits (that cost $ and you have to apply for - a bit intimidating paperwork at first) and there are "first come" permits that are very easy to get. Almost all Forest Service areas have offices near where you will want to go. Walk in, get a permit (you will have to show ID to get the annual fire permit the first time you get a specific trip permit). They give you a short "Leave no Trace" lecture and fire safety lecture. Certain trails have "quotas", others do not. If you choose a no-quota trail, you will always get a permit. If you choose a quota trail first-come permit just be sure to have a few other options so if you do not get your initial request you have the maps you need to do the alternatives. Also, each forest has a web site - and may list the remaining reserved slots. If there is a reserved slot remaining, chances are there will be plenty of "first come" permits available.
For your first backpack, I would NOT recommend an off-trail route. A trail route is better - you certainly can get half mile or more off the trail to set up camp and have solitude. If you are "self teaching" yourself backpacking, you need to start out easy and incrementally increase difficulty while you gain experience. Definitely DO NOT go off-trail until you have good navigational skills (this means read a map - not just follow a GPS).
Also, how far are you willing to drive? An easy overnight starter trip would be the trail up the Mokulmne River from Salt Springs Reservoir. Not many people, no permit hassles (in fact I do this without a permit). It is relatively short (12-miles round trip). There are two roads to Salt Springs Reservoir - the shortest from Hams Station - drop on paved road to the reservoir - and if this is not open due to snow, you can take a long dirt-road up the river.