The San Jacinto Wilderness (the Idyllwild side) is under two hours drive from downtown San Diego, less if you're in North County. Tom Harrison Maps has an excellent map. It's higher elevation, so definitely winter conditions will prevail during the cold months.
The San Gorgonio Wilderness is a bit further, maybe another 30 minutes to get to the Barton Flats area. Again Tom Harrison Maps has a generally excellent map.
Those are the two most montane to alpine, Sierra like areas that I know of in Southern California, but both will be snowy in winter.
For non-snow travel, Anza-Borrego is a good one.