I've written a guidebook entirely devoted to mainland Belize, and also a Fodor's and another Random House guide, and my bias is that the more time you can spend on the mainland the better. Day trips from San Pedro are fine, but at best they only offer a a glimpse at the tip of the iceberg.
The cayes, especially Ambergris and Caulker, are great but in some ways they are not that different from small, less-developed islands elsewhere on the Caribbean Coast or in the main Caribbean.
By contrast, the mainland has many remarkable things to see and do, from the longest cave systems in Central America to incredible Maya ruins (including some where you and the caretaker will be the only people there) to scenic drives through the Maya Mountains to great birding and wildlife spotting to ... well, the list just goes on.
For day trips, I personally think the Lamanai tour offers the most. The Lamanai ruins are remarkable, and you'll see howler monkeys and other wildlife, plus the boat trip up the New River and New River Lagoon is interesting. But it does make for a long day.
Altun Ha is also a good trip.
The Belize Zoo is a must and is really the only place where you will be able to see a jaguar (jaguars are nocturnal and to even have a chance to see one in the wild you'll have to visit areas that are not practical to visit on a day trip).
The cave tubing trips, typically to Jaguar Paw, are also good.
Keep in mind that for the more adventurous types you can do your own trips. Fly or take a ferry to Belize City and then rent a car.
But better still is to spend at least a few days at a jungle lodge in Cayo or elsewhere.
Belize First Magazine