The premier shore dive in San Diego - arguably SoCal - is LaJolla Cove. It's part of the San Diego - La Jolla Underwater Park and is a protected marine reserve. There are multiple dive options including access to the Kelp Beds but they're a pretty good swim out.
The only issue there is usually parking so get there early. >7AM on weekends. La Jolla is a pricey area and popular with tourists so fills up fast - esp. on weekends. Start from the north and drive along the cove, if you see a spot - grab it - you may not find another. There's also a parking garage up the hill in La Jolla, the steps on the side lead right down to the Cove. We usually drop someone/gear at the Cove then go find parking.
LaJolla Shores is another option slightly north of there, lots of classes held there, big parking lot. It can be bleak and featureless but there's a submarine canyon that's an easy kick out and you frequently find something interesting there - big pelagics wander through regularly.
Marine Room is another good dive, often there's leopard sharks in the shallows there. Except for one shallower dive in Mission Bay, just about all the shore diving in San Diego is in the La Jolla area or points farther north.
All the info you need, dive options, descriptions of all nearby San Diego shore dives, shops for tank rentals etc. can be found here:
Divebums - A San Diego Dive Website
You can also access the Kelp Beds via a boat dive to the Point Loma kelp beds which is easier access. Good charters for that are Lois Ann, Waterhorse Charters or Marissa. There's also wreck alley and most boats also go to the sea lion colony off Los Coronados a couple hours south regularly. There's a list on the Divebums site or the bottom of californiadiveboats.com