There is no easy answer. Your rifle may or may not like a certain bullet weight or load by brand X. 150 gr. lead free is a very good starting point and that is where I would start. Unfortunately, in Ca availability of ammo is going to prevent you from trying a lot of different weights and or brands. My first choice as others have said would be Barnes, E-tip or Federal trophy copper. I personally shoot Federal trophy copper the most because it is the most accurate in my preferred rifle. Another rifle prefers Barnes or E-tips. The cheap Winchester 150 gr. work well in a 30-06 I occasionally shoot, and it doesn't like any other factory load. You can spend way more money trying ammo than you spent on the rifle. Hopefully the first box or two you buy will be something your gun will shoot accurately. I will let others tell you about breaking in a new rifle. Some do and some don't, but you should at least read about it. For that you might buy the cheapest load you can since you do not have to shoot expensive copper at paper targets yet.
As far as sight in distance I prefer a 200 yard zero for open Sierra type hunting. In my experience the majority of my opportunities are around 200 yards and any animal less than 200 is not going to be a problem. Research the trajectory of your selected load and decide before you go hunting how far away you feel comfortable shooting a big game animal. Just an FYI, I now practice way more at 200 yards than any other distance. I used to shoot a lot more at 100.
I have a few different rifles and none of them are a 308 but I do like the 308 a lot, just never purchased one.
