Long Beach and San Pedro will be your shortest distance... not familiar with the public launches there. There is however a public launch in Marina del Rey. It's a longer distance (about 48 miles) to Avalon than the approximate 25 miles from Long Beach. If you do use Marina del Rey you could consider starting your trip at Two Harbors (38 miles) and circumnavigating the island clockwise. The trip back to MdR makes for a nice reach. Last trip we were able to fly the spinnaker on a port tack the entire trip home (wind was blowing at 15-20 and made for a quick and exciting return!).
Note that Avalon fills up fast on the weekends so I would plan to arrive on a Monday through Wednesday (once you are there and have a mooring you should be in good shape). Will you have a dinghy or some other means to get to shore? If not, you will be relegated to Two Harbors and Avalon as they have shore boat service. If you do, then definitely plan to spend a night or two in Little Harbor. There is only anchorage there for about 3 boats (bow and stern anchoring is recommended and is how both boats were set up in the pic below) but absent a few campers, you'll have some amazing trails, beaches, and views all to yourself. The pic below is from Little Harbor, 4 weeks ago.
One thing that I'd recommend in Avalon - make a reservation for a couple of lounges at the Descanso beach club (go to Visit Catalina Island - Hotels, Packages, Tours, and more.
). It's great for some pure r&r resort style. No crowds, nice chaise lounges, and drink and food service. Truly feels like you're at an exclusive little Mexican resort which makes for a nice retreat from some of the foot traffice you'll experience in Avalon.