I recently moved to Cyprus. I'm not an expert by any stretch of the imagination, and hauling around a toddler means that we don't do any heavy duty sightseeing. But here's a little feedback anyway.
May I ask why you chose Cyprus as your destination? Fun in the sun? Try to avoid the hottest months (July and August). Or wear lots of sunscreen and rash guards for the kids on the beach! Personally I try to do errands in the mornings and hide inside during the afternoons.
Shopping? If you're shopping on the US dollar, you probably won't want to do any big shopping in Cyprus. Prices are high anyways because of being an island - add in the strong Euro/weak dollar - things can get pricey.
Also, except in the designated tourist areas, stores are closed on Sundays and Wednesday afternoons and forget about evening shopping - stores close 7:30 or 8. (Except convenience stores called kiosks, they are allowed to be open more hours.) Pharmacies have restricted hours too, and you'll want to talk to your hotel reception about which pharmacy is open should you have a need (or else you'll have a heck of a time finding an open one!)
Definitely try a meze meal at a local taverna. It's pricey but worth it for a relaxing (long) lunch.
Any of your family allergic to cats? They are everywhere around here.
Oh, and the local tradition in much of Cyprus is that you don't flush toilet paper down the toilet. There are trash bins next to the toilets that you are supposed to use. I have heard 2 different theories for this, they probably are both correct depending on the location: 1) The plumbing is jagged on the inside, thus paper would get caught and would back up the system. 2) Some places on septic systems and they don't want to pay to get the tank emptied any sooner than they have to.