Taking a taxi to transport you from A to B in Mykonos is certainly a viable option, albeit a costly one. You will find the three main taxi stands in Chora (at the central square of Mykonos Town, right next to the statue of Manto Mavrogenous), at the New Port and at the International Airport of the island. Taxis are one of the most relaxing means of transportation, but also the most expensive by far, as their cost meter is proportional to the distance they will travel.
You can find taxis in Mykonos Town (the square is in front of the OTE building), but also in Fabrica, on the other side of the city (where the bus stops for the southern beaches). You can call taxis 24 hours a day at the following numbers: 22890 23700, 22890 22400. Sure enough, taxis are one of the easiest ways to travel around, but you will find out that booking luxury transfers in advance will personalize your experience and prove to be far cheaper overall.