My own preference is for Gozo, it is much more laid back, quieter and prettier. I could recommend Xlendi or Marselforn.
The ferry crossing is from Cirkewwa in Malta to Mgarr in Gozo and takes about 20 minutes. Gozo is a very small island, only a few miles wide but there is so much to see and explore on there.
Malta, whilst I love Malta, places like Bugibba and Qawra are where the main tourist areas are..and they can be quite lively at night (and also very tacky and run down). Melliha Bay is also popular but I would recommend all of these places for the younger person. I think for more mature people, or perhaps people like me who prefer things quieter..Sliema is very nice and from there it is easy to go all around the island.
Malta has a very good bus service and it is very cheap to get around. You could quite easily visit Mostar, Mdina and Valetta etc where there are many things to do/see/visit.
Malta does not have many beaches in the traditional sense as most are rocky so if you wanted to sunbathe, it is more likely to be actually at your hotel.
http://www.google.com/url?sa=t&source=web&cd=4&sqi=2&ved=0CFIQoAIwAw&url=http%3A%2F%2Fwww.victoriahotel.com%2F&rct=j&q=sliema%20hotels&ei=sjCkTaC_K9SbhQfezaDACQ&usg=AFQjCNF-FjYhCSR_R6gZhhoaLJ-yO1fnvQ&cad=rja
http://www.google.com/url?sa=t&source=web&cd=1&sqi=2&ved=0CDsQoAIwAA&url=http%3A%2F%2Fwww.excelsior.com.mt%2F&rct=j&q=sliema%20hotels&ei=ATGkTd6EAoG2hAfkgMnMCQ&usg=AFQjCNEVZPfGpSFop5Jyyy_5lIdY7Mm4Cw&cad=rja
these are sort of more in the higher bracket..
http://www.sovereignhotels.com.mt/
I am not really sure what type of accomodation you are looking for.
Just let me know if you need any more info and I will see what I can come up with