Famagusta lookout point is drivable and you can park up and look across etc.
Maps if you need one are all over the place, probably get one in the back of your welcome pack.
Don't miss going on a jeep safari up the Troodos mountains, see the sights, get a meze lunch, sample the wines....
Don't miss the sunblock and water !
Don't miss a Traditional Cypriot night with dancing and music, food and wine.
Hire a car and visit places like Kavo Greco as the scenery is stunning, think of a beautiful blue sea and and the rocks leading to them as a painting, well thats somewhere near.
The jet boat thing in Paphos harbour is different and a bit thrill seek'y
Eat the local food as some of the dishes are better than you could possibly imagine. Also try some of the other types of food like Italian, Indian, Mexican, Japanese, Chinese if thats your bag, but choose wisely, look for authentic (like Nautilus in Pernera for real Italian food).
thats a few things to get you going...
Steve