Tikal in Guatemala's Peten of course is a must-see. It is large, many of the buildings have been excavated and the setting in the jungle is amazing.
I've visited all the major ruins in Belize and would list them this way, in order of probable interest to the archeological non-professional:
1. Caracol in the Chiquibul wilderness beyond the Mountain Pine Ridge -- although this site is actually larger than Tikal, excavation is so far limited, but the setting is wonderful and the trip there by road is extraordinary.
2. Lamanai -- the setting on the New River Lagoon is beautiful, and as at Tikal you'll see howler monkeys. This makes a great day trip from San Pedro.
3. Xunantunich -- easy to get to and, though small, very interesting, with an excellent visitors center.
4. Altun Ha -- also fairly easy to get to, and a popular tour from San Pedro. Small, but you can see the temple that's on the Belikin beer bottle.
5. Lubantuun near PG -- very small but interesting site, and the present day Maya who live nearby are special.
Also highly worthwhile are trips to the caves (some on private land) that have Maya artifacts, such as CheChem Ha, Flour Camp, Barton Creek, etc.
Cahal Pech, El Pilar, Santa Rita, Cerros, Nim Il Punit, La Milpa, Cuello and other sites are all worth visiting if you are in the areas where they are, but I don't know that I would make a special trip unless I had a keen interest in Maya archeology.
Belize First Magazine http://www.turq.com/belizefirst/
P.S. My new book, Guide to Mainland Belize, covers all these sites in detail (shameless plug: it's available from Amazon.com).