I'd recommend Lamanai and Caracol.
There is no public transportation directly to the Lamanai site, but public buses (running between Belize City and Corozal) can drop you off on the highway where the tour boats wait -- just tell the driver or conductor where you want off. Most tours begin around 9 or 10 am and start with a trip along the river to get to the site.
Altun Ha is a small site. Buses run infrequently to villages near Altun Ha, and then you'd need a taxi to complete the trip. Tours are an option.
For Caracol, Xunantunich and Cahal Pech, I'd suggest staying in San Ignacio because the town has a good selection of accommodation for all budgets.
There is no public transportation to Caracol. Visit the tour companies in San Ignacio and join a tour. Also visit the Cayo Welcome Centre for tourist information.
You can get to Xunantunich by bus/foot. Get a bus from SI to Benque and ask to be dropped off at the turn-off to Xunantunich -- that should cost you BZ$1 or $2. After crossing the river on the small ferry, walk the 1 or 2 km to the site. A few hours is all you need here. After walking back to the highway, wave you arm up & down at an approaching bus to get it to stop.
From the centre of San Ignacio, you can easily walk to Cahal Pech.