Re: Southern California, "May gray - June gloom"???
Ho parenti che vivono in Calabria. Ci sono onde li, no?
I've grown up in San Diego and travel to Italy from time to time as well. The others have explained the gloom pretty well. It's not very pretty, but it tends to keep the wind down. Personally, I hate the grey. You'll have fun in California.
You can plan your trip by keeping an eye on swell direction. It sounds like you are doing plenty of research and that's your best bet. In May and June, you'll likely want to hit spots that get a southern swell such as Malibu, Newport Beach, Huntington Beach. Hit the classics like Rincon and Trestles. Those places will be a bit crowded on good days and will have some really good surfers to contend with, but you should be fine.
The biggest issue you will have in California is the crowds. Some localization, but not like Hawaii. There will be fewer crowds in northern Cal and the Central Coast should be the best. Lots of surfers throughout LA (Malibu will be good for southern swell) and San Diego. In San Diego, your best waves are going to be in Oceanside and Blacks Beach in La Jolla, maybe Sunset Cliffs. WindnSea in La Jolla should be good as well, but beware of localism in that spot.
Bring a good wet suit. You'll need a full in the north and central coast. There it can be sweltering hot a few miles inland and 30 degrees colder at the coast. By the time you hit LA, you can get away with a shorty. San Diego will be the warmest.
Mexico - I don't know if your visa will allow you to visit Mex, but there may be some great surf just south of the border in Baja, Calofornia. Sometimes I cross the border for a morning surf session. Find a place called Baja Malibu or K38. Those are the best. If it's flat, head to La Fonda -it's always got waves. And don't leave without eating a bunch of street tacos. Don't drink the water.