Hi Sweetheart,
Pernera is right beside Protaras, it's kind of a suburb of it. I'd say Protaras or Pernera would be best for you.
Sandy beachfront on both resorts (the two are connected by a boardwalk which is lit up at night)
Here's a map of Protaras and Pernera just click on it to enlarge it.
Nissi Bay is central Ayia Napa, it's the beach road area, it would be quiet enough for you and the hotels are lovely. It's a good 20 mins to the noise and action.
Fig Tree Bay is Protaras' main beach.
The Hala Sultan Tekke Shrine/Mosque is in Larnaca, near the salt lake beside the airport, I never went there. You can't consume the salt on the saltlake as it's polluted by the airport, though flocks of flamingos do visit there in March and April.
If you want a day out, hire a car and drive to the mountains. (there would be bus trips too, but as Cyprus drives on the same side of the road as we do, so it's very easy to drive there)
The village of Lefkara is in the foothills of the Troodos mountains, worth having a look at the Cypriot ladies making lace. There's also fabulous fish restaurants and water mill at Kakopetria on the way up to the mountain summi, the top is called Mount Olympus, and a lovely village called Omodhos on the way down to visit if you do hire a car, just for one day.
You could also drive along the "green line" which divides Northern (Turkish occupied) and Southern Cyprus (Greek Cypriot) this is VERY interesting.
What board basis are you after? I could reccommend a hotel / apartment if you like? Tell me what dates too, and I'll tell you where will be busy / where will be cheaper.