When you get off the tender from the ship you are basically in the middle of Belize City. When you get off the tender you are in the shopping area, which is closed off from the general public. It's a fairly small place with the usual tee shirts, bags, leather, wood carvings, jewelry stores, etc. A couple of liqour stores and I remember at least one pharmacy. I was there last week and there was band set up in the shopping area. I don't know if this is a regular thing or not. There are a couple of bars & restaraunts. If you go through the shopping / terminal bldg. to where you get on the tour buses, there are a few small stalls set up across the street.
If you are interested in beaches, I would closely check the shore excursions offered by your cruise line to see what they have to offer. Belize City is not somewhere you would want to go off wandering around on your own. The tender ride is 4-5 miles in a high speed boat. It takes 12-15 minutes to and from the ship.
Belize is a pretty country when you get out into the countryside but if you do take a tour on your own, keep an eye on the time as Belize City is not the place you would want to miss the ship.