Stanford University - You can't drive around Stanford without being cut off by bikes, so why not ride one? And if you want a good ride while you're in the area, do "the loop," which is Alpine Rd to Portola Rd to Woodside Dr.
University Ave - I'd avoid this on a bike because it's always ridiculously busy. However, Palo Alto has lots of casual suburbia rides, and note that Bryant St is a "bicycle boulevard" -- lots of it is usually car-free except for the locals, and it'll take you to Charleston, which runs into the Google campus.
Sand Hill Road - A good ride with some reasonable hills. Take this road south to Portola and then try the gold standard of climbs -- Old La Honda Rd. If you can make it from Portola to the stop sign in less than 30 minutes, you're in good shape. Less than 18 and you're Lance Armstrong. Don't go down Old La Honda, though -- go north on Skyline a bit and take 84 down since the visibility and view are better.
Castro Street - This place really wants to be a hip, urban center, but believing whether it is is up to you. Bike here and, like PG said, stop at Red Rock. Or grab a gyro and some baklava from the Gyro House, which is terribly underrated.
Google - The Google campus is so bike accessible that they provide clown-like bikes for employees to ride around. Hit up the main campus and see the dinosaur and, currently, granite head sculpture exhibit. Of course, you should probably show up with someone who works there and can throw in a free lunch.
North of Google - If you're biking from Palo Alto to Google or vice-versa, cut north through the Shoreline Lake reserve and the Palo Alto Baylands Nature Preserve. Keep your mouth closed during the spring months so you don't inhale a cloud of gnats.
Skyline Drive - A mixed bag. In the morning this is beautiful, and when the fog clears you can see the entire valley. It's epic. Unfortunately, the road is used as a testing ground for Porsche and motorcycle lovers -- people have died, and I have more than one friend who was almost hit.