I believe if you cut over at Half Moon Bay, you'll get plenty of what you're looking for. Starting at Santa Cruz doesn't make sense for two reasons: one, you'll miss most of coast because 1 veers away from the coast for a while as you go from Santa Cruz to Moss Landing and again until just north of Monterey, and 2) the SC-Monterey stretch is nice, but not the prettiest part of the drive. Are you looking for anything particular to do in Monterey, or do you have that palnned out? By the way, I grew up "next door" to you, in Highland Park!
PGdenizen has given great advise. A couple of things to add:
The road you want to take is Highway 92 to Half Moon Bay.
It will get scenic as soon as you get to Highway 92 and it passes over the hills.
Do recommend once you get to Half Moon Bay to drive a few miles north to Pillar Point Harbor. It will be a few minutes drive.
Then drive south on 1....(Note: Driving through downtown Half Moon Bay there are a few arts and crafts stores now that are interesting...This is not on the direct Highway 1 route is something you might also consider doing.)
Continuing down the coast, there will be various beach areas.
In particular if you want to see elephant seals, Ano Nueva has a walk out to elephant seals.
Also along this route is Pigeon Point Lighthouse:
You can continue down Highway 1 to Santa Cruz. There are a few attractions that you will want to consider driving to when you get to Santa Cruz that are not directly on Highway 1.
Santa Cruz Boardwalk:
Santa Cruz Wharf (good place to stop for and also see seals at the very end of the wharf).
(I'd recommend driving to the wharf even if you don't spend anytime at the boardwalk - note the boardwalk is free; the rides cost so you can see the area relatively inexpensively.)
Then continue south on Highway 1. There are verious beach parks not directly on Highway 1 such as New Brighton State Beach. www.parks.ca.gov/default.asp?page_id=542
Again - not direclty on the road, but worth a stop if you want to walk along the ocean.
From there continue south till you get to Moss Landing. You will see a big power plant here...a marina; and there is another state beach here.
Also close by to this area is Elkhorn Slough:
After this continue south on Highway 1 and Monterey.
Highlights around Monterey are aquarium; 17 mile drive; Pacific Grove area by Asilomar State Park; and if possible Pt. Lobos State Reserve. http://pt-lobos.parks.state.ca.us/
If you're set on seeing part of Big Sur you can continue south and drive as far as Pfeiffer Big Sur State Park and then return.
So - my answer in short is: a little north of Half Moon Bay and then decide where you want to stop and see more of during the day. On the way back, recommend taking Highway 101 back to SFO area as it will be faster. Have fun.
I am referring you to a post by "paradisetreker" that I found about a month ago. It is so marvelously detailed. I note you have only one day, but you may find it helpful to highlight in points along the way that would fit your schedule.