Given the nature of the trips you might want to consider spending a few days on the mainland especially for a trip to Tikal as getting there from San Pedro requires a flight there and back and will really limit your time to enjoy this enormous and spectacular Maya site in Guatemala.
Community Baboon Sanctuary is about a one hour drive from Belize City. I believe Tanisha Tours offers a day trip there from San Pedro along with a visit to the Maya Altun Ha site. I'm not sure if you'd want to squeeze in a trip to ruins on the same day though. You can get information here. http://tanishatours.com/.
Tanisha also do a very popular trip to the Maya Lamanai site too as do many other very reputable tour companies.
If you decide to do them all from San Pedro, keep in mind that they are all long day trips.
San Ignacio is a good base for the Tikal trip and you will find it a lot nicer than Belize City. San Ignacio is much closer to the border and many good tour companies do trips from there to Tikal. San Ignacio is also a lovely town and will give you the opportunity to enjoy so much more of Belize. You could then hit the Baboon Sanctuary on your way to San Pedro. The Lamanai trip from San Pedro is worth doing because the river boat ride is such an amazing part of the experience.
Hope this gives you some ideas to get started.