Okay, first of all, welcome to California.  You will be surprised at the hunting that's available here.

Deer, elk, antelopes, and sheep are controlled by a lottery system here, which the deadline to apply is June 2nd.  Bear, pigs, turkey, are all over the counter.  Mountain lions can not be hunted at all.

Depending on where you put in for deer, it can take 8-10 years to draw, or you can find "zones" that actually have surplus tags and are given out over the counter.  As a basic rule of thumb, the further south you go, the more permits available and the success rate is lower.  I hunt in San Diego, and our tags, both for rifle and archery, can be bought over the counter.

Howeve, our muzzleloader and doe permits go out in the lottery.

As for elk, there are not alot of permits for that, and the tags are very hard to come by.  There are always some reserved for a drawing that is not based on points, but in some cases, that's 1 tag per zone.

I have not put in for antelope, but I know people who have had success doing so.  The sheep tags are almost impossible, because there are so few state wide.  The closest one to me usually gives out 1 tag, if it even is an open season.

Pig hunting is great in the middle to northern parts of the state.  Lots of private land hunting, if you want to pay $500 to $1000.  Public land holds pigs, you just gotta know where to find them.  If you get a job with the forest service, you might be able to do a little scouting "on the job" .  tags for pigs are unlimited.

Bear tags can be bought over the counter, and the hutning is decent for them.  The season is funny though, in that it goes on a quota.  It will open on a certain date, but you need to continually check the website to see if it's still open.  They will cut it off when a pre-deermined number of bears are harvested.  I believe it's around 3,000 bears.

Finally, dove hunting is a huge tradition out here, for the opener.  Also, there is very good quail and pheasant hunting, and the waterfowl can be really good or really bad, depending where you are.  You will just buy an $8 stamp to go on your regular license, which will allow you to shoot all this type of "Upland Game".  That includes turkey, which you are allowed 3 birds in most part of the state.

Anyway, feel free to shoot me a PM if you have any more questions.  I'd be glad to answer them for you.