You may not have as much trouble as you think. The only places where you will have a REAL problem getting reservations for hiking or back-packing are Half Dome in Yosemite and Mount Whitney in Inyo National Forest. You will need a permit for many other places, but a party of only two people will have little trouble getting a permit to somewhere of interest. Just discuss your hopes, experience, and equipment with the ranger at the permit station (wherever it may be); and you'll get good advice on which place with openings (most places OTHER than the above two invariably have plenty) bets meets your needs. If your top choice has already reached its quota, then another trail, just as good, will almost certainly be available.

This URL will give you an idea of where you will need and can get permits in advance
http://www.recreation.gov/welcome.do?topTabIndex=Home
1) For the pull-down menu below "Interested in," select "Permits and Wilderness"
2) In the box below "Where?", type out the first word of where you want to hike or back-pack.
3) Click on the place (a few will quickly pop up) that is exactly your place of interest.
4) Click on the [Search] Button.

If you can get a permit in advance, your choices will be displayed; at which point you can check available dates. If no choices for hiking permits are displayed, then you almost certainly won't have any trouble getting one when you arrive.

Yosemite is a must to include in your adventure. It is one of the most beautiful places on Earth and has great hiking trails. Plan to spend 2- 4 days . Make reservations a year in advance or as soon as possible.

Regarding other biotopes:

Oregon State University and Humboldt State University are famous for their oceanography. OSU has a an excellent museum in Newport Oregon; it addresses both habitat and things like wave technology. Newport also has a separate excellent aquarium with a well-known shark exhibit. Not sure what Humboldt has available to the public. Monterey Aquarium in California is justifiably famous. Tide pools are available many places along the Northern California and Oregon coasts.

Thanks for the tips!

Any advice on camping in Big Sur? Is it necessary to book a tent site in advance (going thu-fri 14th-15th June)? What are the best (peaceful) campsites there?

Yes, I would recommend advance reservations.

Big Sur camping in Pfeiffer State Park...check out the current conditions due to renovations. Still might be worth staying there.
Don't forget warm clothing, as you will encounter fog in the morning most times, and sometimes all day.

In the mountains, pay attention to the directions on keeping food stored safely away from bears. That includes toothpaste, lotions, sunscreens, and hidden candy bars you forgot in your pockets. Those bears are trained professionals, including skills in prying cars open with their claws.

It sounds like you have a wonderful trip

I just noticed how long your trip is.

My guess is that while you want to do backpacking, on occasion you will want a campsite where there are showers and a place to clean-up. Be aware that actual campsites on weekend days should be reserved ahead of time.

I wanted to explain a little more about the campground differences. First, you can reserve at almost all of these systems online.

The federal campgrounds are the most rustic.

The state campgrounds tend to be the best mix of decent showers and still beautiful locations. Some state systems have certain campgrounds that are first-come first-serve, so there are no reservations. In some cases they are walk-in from your car and more rustic. That might be something to check out as well.

Note that if you cannot get a state campground, there are also KOA's (Kampground of America) and Thousand Trails, two private chains where you will pay more per site; one advantage of these is that many if not all have washing machines and dryers for your laundry; you put coins in. They often also have a swim pool or hot tub, and maybe even a small restaurant. They are "camping light" but might be easier than finding a separate laundromat, which in this country can be hard to find and not always the most pleasant locales when you do.

Finally, there are also independent campgrounds. They also tend to be more for people in RVs and trailers, rather than tents. My in-laws were full-time RVers for a few years, and they tell me these independents are very, very uneven in quality. Some are absolutely wonderful with amenities, but some were people who could not afford a home, the sites had lots of clutter and junk and felt unsafe. There are websites devoted to RVing and camping where people rate these but in general I'd say avoid them.

Also, regarding firewood: Many states have been having a problem with incursion of beetles from other places. The philosophy among responsible campers these days is to buy firewood at the campground you are visiting but please do NOT take it with you to the next location. Leave it for the next person using your site and if you're lucky someone will leave some for you when you arrive elsewhere.