I went to Malta a few years ago and loved it. A week would be better, but you can see a few things in a few days.
We were lucky and were self catering in St Pauls Bay. Outside the complex was a bus stop which took us to Valetta (the capital) and from there, almost anywhere in the island. We didn't need taxis or to hire a car.
I recommend the Catacombs of St Paul and St Agatha (next to each other),
Mdina - the Silent City (although not so silent as there were lots of road works going on when we were there!)
Valetta - the Capital - for browsing the shops, but if you are interested in WW2, there is the Malta Experience,/ National War Museum, and various churches and cathedrals.
If you are interested in Archaeology, there are plenty of small sites. I wouldn't bother with Gozo on such a short trip.
If you like bread and butter, they make the best!