We highly recommend that you check out the Aquarium of the Pacific, the largest aquarium in California. We love the penguin exhibit but they're all great. You can also rent swan boats nearby in Rainbow Harbor.
If you have time to spend a day or two on Catalina Island, it's well worth the ferry trip from either Long Beach or San Pedro. You can go snorkeling or ziplining in Avalon or have cocktails on the beach, or go hiking in Two Harbors on the other side of the island.
Right by the Aquarium there are several agencies offering whale watching tours. We haven't been yet but it's supposed to be a ton of fun.
This retired ocean liner was built in the 1930s and is now a hotel and museum, available for guided tours year-round.
Retro Row on 4th Street has a ton of great restaurants, kitschy vintage shops, and our favorite movie theater, the Art Theatre of Long Beach.
If you have access to a car, there are some incredible cliffside views in nearby Rancho Palos Verdes. Places to check out include Point Vicente Lighthouse, Terranea Resort, and Founders Park.