Taxis are not really popular in Ragusa as almost everyone has a car or a scooter, more popular with the young population. So there is no public taxi network.
You can however find some private cabs by going to Piazza Liberta' where there is a taxi stand, or try by looking on the phone directory or the yellow pages.
These taxis can also bring to the airport should you prefer a car to going by bus, but I'm not sure about the tariff, could be expensive.
Getting to Ragusa might not be as easy as getting to other major towns. If you arrived in Sicily at Catania Airport, the chepest way to get to Ragusa is by bus. You can take it straight from the airport.
If you don't have a car the easiest way to get around Ragusa is by bus. The recently renovated fleet covers most of the town including going to and from the old Ragusa Ibla.
But, as far as I know, buses don't operate in the night.