You can rent a car on the island. However, unless you are used to driving on the left side of the road and can handle unpredictable driving, you would be better off with a taxi or bus. A good map is a must. Petrol, as almost everywhere, will seem expensive by US standards.
By bus
One of Malta's joys (at least in small doses) is the wonderfully antiquated public bus system, consisting of 1950s-era exports from England usually kitted up with more chintz than a Christmas tree plus icons of every saint in the Bible and then some. Fares are very cheap and even the longest ride across the island costs less than a pound; the only catch is that almost all buses radiate out from Valletta, so you may have to detour back to the capital to reach your next destination.
It is always best to pre book your car rental online as this works out a lot cheaper than booking when you arrive. You can compare prices of local car rental companies for malta.. According to the mediterranean markets, malta has very low rates for car rental. Any driver and additional drivers must take with them there driving licenses in order to be covered for by the insurances provided by the local car rental supplier.
By taxi
Malta's taxis are a ravenous lot and fares are quite expensive. There are black taxis, which have to be called in advance, and white taxis, which can pick you up off the street. Both have meters that are uniformly ignored, figure on LM5 for short hops and not much more than LM10 for a trip across the island. Also note that plenty of black taxis break the rule when nobody's looking, and this is the cheapest way to hitch a cab ride. All taxis must have the third letter of their licence plate with letter Y otherwise these aren't authorised taxis and might not be very safe.
By ferry
There is the regular ferry service between Cirkewwa on Malta and Mgarr on Gozo, it goes every 45 minutes in the summer and almost as often in the winter, The trip there is free, but to go back to Malta costs 2 Lira. There are also irregular services to Comino.