Full day, the zoo and Balboa Park. It's a beautiful zoo and just feels like San Diego. Balboa Park, where the zoo is located, is a wonderful park filled with museums and places to sit and explore. I end up in this park 5 or 6 days a week. It's a gem. I never get tired of it.
I'd avoid Old Town, tourist trap.
Birch Aquarium is wonderful but small, but it also puts you in La Jolla which is a great little beach town about 12 miles north of Downtown San Diego. Great places to walk, tidepool, some fun playgrounds, and of course, some very nice beaches to play on. This also has the Children's Cove which is now home to a sea lion colony. Great fun to explore!