Buy your liquor at duty free, by the baggage claim at Belize International airport or your beer at Belkin distributor in San Pedro, south end of town.
As for groceries, depending where you are located, purchase supplies in town at Super Buy on Back Street and veggies and fruits directly across the street beside hardware store. If you are staying south of town there is a Super Buy just north of Banyan Bay Resort.
If you like fresh fish, try the guys on Back Street across from Caye Supplies on Tues or Wed am?
On the marina side, behind Back St, fruits and veggies can be had from the boats from the mainland, early Tues. am. Hope this helps