OEX is located at La Jolla Shores. The shop rents out dive gear, teaches classes, and has DMs that give guided tours of the La Jolla Shores/Cove area.
For a novice out-of-town diver, I'm not sure I would recommend diving from a kayak in the La Jolla Shores area. There's plenty to see/do not too far away from shore.
La Jolla Cove can easily be done as a shore dive. Kick straight out to yellow Buoy A and descend on the buoy chain. Keep your eyes open for Giant Sea Bass that tend to hang out in the middle of the water column. Sometimes you'll see harbor seals or sea lions in the kelp in that area. If you're lucky, you'll see topes (soupfin sharks). If you're really lucky, you'll see a sevengill shark. For someone unfamiliar with the dive site, I highly recommend surfacing to see exactly where you are so that you can take a safe heading back into the Cove. Depending on the wave action, if you aren't paying attention, it's possible to get dangerously close to the rocks. The lifeguards aren't thrilled when that happens.
Prior to your trip, join the email list at Divebums.com. You can ask to tag along with a local diver on a shore dive. There are a lot of friendly, helpful local divers here in the San Diego area that would be willing to give you an informal tour.
Vis conditions can be iffy at La Jolla Cove. Generally speaking, vis conditions are better at La Jolla Shores.
There's more kelp at La Jolla Cove, though.
