Don't miss the mosaics. I've seen the Byzantine mosaics of Ravenna (which are truly wonderful) and I have to say that the mosaics of Salamis left as deep of an impression on me (although in a different way). It's impossible to gaze upon these works without considering just how long ago they were produced - it might also occur to you, as it did to me, that seeing such things is an eye-opener in terms of our belief that our culture now represents some kind of magnificent peak of civilisation. Be warned that you have to be a little persistent to find these - they're on the map, but tucked away in one corner of the marked place - don't give up until you've seen them though!
Whether you're staying on the Greek side of the island or the Turkish northern part, you'll need wheels of some sort to get to Salamis.
It lies on the northern side of the Green line. Thus if you want to get to it from the south in the car you will need to go through border procedures - very simple really, you just get a visa for yourself and insurance for the car (12 pounds for three days when we visited). If you have been persuaded by your hire car company not to cross into the wild north then take what they say with a large pinch of salt - the drivers are no worse, you are insured, and security appeared to be no more of an issue than in the Greek part of the island (where your car is 1000% safer than in Britain for example).
Once you cross the border consult your map - Salamis lies north along the coast from Famagusta - it's right on the sea, so as long as you're looking out for the sign then you'll have no problems.
Salamis is largely untouched by tourists and tourism. This means that there are no guards or musueum employees keeping an eye on things as you wander round. It also means that a lot of the antiquities are simply lying around where they have been left or have fallen. Tempting as it might seem to help yourself to a piece of treasure, don't. If everyone who visited took only one small piece of treasure then imagine how quickly the site would empty. Taking things is nothing short of looting - and it is illegal enough to land you in hot water if you get caught. Don't do it.
Favorite thing: Salamis is on the Turkish side of the Green Line. We had a sudden moment of panic at the ticket office when we realised we had no Turkish money with us. Worry not - Cypriot pounds are quite acceptable - you will even be provided with change in Cypriot pounds if you wish.