http://www.parks.ca.gov/parkindex/

I’d strongly suggest not camping outside designated areas; someone will call the cops.

The LA area would give you a lot to do, but you might consider traveling the coastal area between LA and San Francisco (or even further north or south). Lots to see and the geography changes from south to north.
References :

If you want to do backcountry backpacking, California is a great destination. We have the Pacific Crest Trail, the Tahoe Rim Trail, the John Muir Trail, and more. If you are interested, there are lots of books about these trails.

For backpacking in cities, sure, you can stay at hostels. I personally can’t imagine how I would spend 2 months in LA unless I was just going to different restaurants and nightclubs every night, but I’m not really a big fan of that city. If I had 2 months to travel in California, I would try to hit up the cities (including San Diego and San Francisco), the national parks (Yosemite, Redwood, Sequoia, King’s Canyon, Lassen…) and other beautiful places like the Monterey Bay and Lake Tahoe. But that’s just me.

I don’t know about Canada, but in America people take private property pretty seriously. If you are caught camping on private land, and get caught, you could get arrested and/or chased away by a crazed rancher with a shotgun. Even on government land, you almost always need a permit for overnight stays, and could get a citation and fine if caught. Urban areas of California have large homeless populations, so if you sleep on the street or in a city park you might end up on the wrong side of a turf war.

Something else to consider is that California has pretty terrible public transportation. Amtrak and Greyhound will get you to some places, and San Francisco has a decent subway system, but in general you need a car to get around. The link in the answer below is great for finding campsites, but you will need a car to get to most of them. And you mentioned gap year, so I am assuming you are around age 18 – which means it will be near impossible to rent a car. Most car rental companies will only rent to people 25 and older. Some will rent to 18 year olds with steep fees.
References :
Lifelong resident of Northern and Central California