They are everywhere you go. Just off the square, on the main road, outside the clubs.
If you cannot see one then you do not have to wait long for one with the exception of August when the Cypriots turn to tourists too!!
Also, you can always call a taxi from your accommodation’s reception (assuming there is one.) All hotels will work with at least one Taxi provider and will also let you know how long so you can squeeze in a drink before it comes.
Just in case you’re a large family or a group of more than 5 people then there are stretch taxis available.
You can even hire out Limo Taxis for those special occasions or just to enjoy the ride in one!