Belize City gets such a bad rap but it's not as bad as they say. I have had some awesome times there and some people even live there and like it!!!

The main hotels for tourists are the Radisson, The Biltmore, The Great House, D'Nest B&B. The Biltmore & D'Nest are closer to the airport but kind of in the suburbs. The Radisson & Great House are in the Fort George area on the coast. Both have decent restaurants and the Radisson bar is a popular place. The Biltmore bar is OK as well and has live music several nights a week. There is a new place called the Riverside Tavern that has good food and a popular sports bar type atmosphere. Can get there by taxi from any of the hotels. The Princess Hotel has a casino if you are interested in that but don't stay there - just play there!
There are some very cheap hotels in the city as well like Hotel Mopan, BelCove & Seaside Guesthouse but you would probably want to see them first before booking. Cheap in Belize does not mean a Motel 6 type place!

And there are some dangerous areas you would want to avoid (like any city).
That said, Bz City does NOT make a good first impression of the country and it is possible to get to the islands the first day without spending a night there. Last flight is at 5:40. Or you could get a driver/taxi to the inland areas. Possibly can make a bus to Cayo or elswhere if you don't get in too late.