First and foremost, avoid going to the island if Santa Ana conditions (dry Norteasterly) are forecast. There are very few places offering shelter from these conditions, and they can be extremely dangerous.
Coming from Channel Islands Harbor, the closest anchorage on the island is Smugglers Cove. This anchorage is large and popular, but can be roly and uncomfortable at night, especially if there is any swell or weather coming from the south. Nearby is Yellowbanks Anchorage, which can offer better protection from the prevailing NW winds, but less protection from the swells associated with those winds. If you continue onwards around the south side of the island, Coches Prietos is a beautiful cove, and offers some of the best shelter to be found on this side of the island. Below is a picture of the beach at Coches. No parking problems here!
If you turn right at San Pedro point (the east end of Santa Cruz) instead of left, there are also many fine coves and anchorages on the North side of the island that you can get to in one day. Two very popular and sometimes busy ones are Scorpion and Little Scorpion. These each have their quirks, and there is no substitute for the fine book by Brian Fagan called Cruising guide to Southern California's Offshore Islands for learning these things, but for now I'll say, given the choice, pick Little Scorpion. As you go west, you will find Potato Bay, which is a beautiful but not super well protected anchorage from the prevailing Westerly. The next really good cove is Prisoners' Harbor, where the old landing site of the Santa Cruz Island Company is located. The signs of civilization may put some people off, but it is rarely crowded. My personal favorite, and that of many, is Pelican Bay. It is at about the limit of an overnight itinerary, but well worth the trip to get there. Again, you must plan to be sharing with others (maybe many others) on weekends in the Summer.
Stay tuned to this spot for future additions of information. We'll have some suggestions for three and four day trips upcoming.