How long are you there for?
Gozo is tiny and can be done in a day trip to be honest, but it's perfect if you're going just to lounge by the pool/beach and chill out.
Malta is only 20-30 minutes away by ferry (the following are all on Malta).
Sliema, St. Julians and Valletta are the key urban areas with the best restaurants, bars and clubs (Paceville is the area for the nightlife and technically in St. Julians). Valletta is incredible for it's architecture and history. Check out Portomaso for some really nice, high end (but pricey restaurants). IMO one of the best restaurants in the entire country is Ta'Xbiex Waterpolo Club (and it's very reasonably priced for how good it is) - the Frito Misto starter is superb. It's perfect for lunch or dinner. D'Office in Valletta is where I always used to go eat lunch when I was there as well. Reasonably priced excellent traditional Maltese/Italian cuisine in a really nice homey atmosphere.
Check out Rivareno in St. Julians for ice cream. It's some of the best out there, simply amazing.
The promenade from the top of Sliema (Tigne Point - that's also where most of the shopping is) that goes all the way along to St. Julian's/Sliema is an amazing walk with really lovely seaviews and nice restaurants. Cafe Il-Gabbana is along the promenade and offers some lovely food and drinks for a little of a respite
Marsaxlokk and Marsascala are fun little traditional Maltese fishing towns worth visiting for the seafood but there's really not an awful lot there; something to see though.
Mdina is definitely worth seeing. It's the old capital and like Valletta has these huge, ancient walls and fortifications which are super cool to see. Nice views of the surrounding countryside up there as well. Valletta and Mdina are famous for their fortifications because they're so well-preserved that loads of movies like Gladiator, Troy and Kingdom of Heaven were filmed there to make use of those walls.
You'll have a great time and the weather will be amazing for you.