Hi 12 noon,
As M&N has said all the best beaches are at the north end of the Island. Although Mellieha is a relatively large beach there are plenty of smaller ones such as Armier & Little Armier which are worth visiting but you probably would need to use a car to get to them. Golden Bay is also a very popular beach and fairly easy to get to with the regular bus service. There is also the new Raddison Golden Bay 5star hotel, a bit pricey, but it is right on the beach. Gozo also has some nice beaches especially Ramla beach.
Although folks often say that Malta's not the place to go to for beaches I do dispute that. I personally don't like just lying on a beach all day and have always found Malta to be is a good destination to explore though I do recommend you hire a car if you want to get to some of the out of the way places. Failing that the buses are cheap to use. We always tend to by a weekly ticket so that we're not looking for change all the time and are able to use the buses to go wherever we choose,
Trev