I'm pretty sure supershuttle won't go this far. I'm not aware of any other shuttle services.
To go by train, you'll need first to get from the airport to Sante Fe Depot. There is a bus that picks up at the terminals and drops off at the train station every 15 minutes or so -- it's the 992 bus. It costs, about $2.25 per person one way, so depending on how many people you have, it might be more cost effective to take a cab (particularly if you have lots of luggage). That would be $10 to $15. I think that some San Diego cabs are only allowed to take 4 passengers, so you might need to take two cabs to the train station if you have more in your party.
Amtrak to Anaheim would be about $22 per person. I think there are about 5 to 7 trains per day. It's also a glorious ride -- through Del Mar you go right over the bluffs over the pacific. Sit on the left side of the train on the upper deck! Unfortunately, not every train will have baggage check and claim service in Anaheim, so you have to check that and make sure you know in advance. Frankly, if you can manage your bags yourself, you're better off that way anyway. Amtrak luggage service on shorter routes isn't exactly a well oiled machine. The checked bag dimensions for the train are also different from airlines dimensions, so again look into that if you'd be planning to check bags on the train.
Once at the Amtrak station in Anaheim, you can take a very inconvenient city bus or pay about $10 to $15 for a cab to DLR.
If your party is large, you might want to just call a cab company to see what they'd quote you.