It's always worth having enough cash for food - a lot of places off the beaten track don't take cards. Cypriot coffee is very strong, thick shots of coffee, so that's what to go for if you feel the need for some caffeine. They'll ask if you want with sugar or without.
Mobile data is usually excellent all over Cyprus, and you can use Waze or Google Maps to get around by car very easily. Don't get too big a car if you're driving around the mountains, the roads are thin and windy and you need something nimble!
Cyprus has local wine which is generally excellent, always choose a Xinisteri over any French grape white. Reds are similar. The local fortified wine (Commandaria) and spirit (Zinavia) can also be very good (and dangerous in the latter case!)
Massive plates of fish, kids play area, overlooking the sea in a little protected stone beach.
Really good, home cooked food. Probably the best Cypriot food I've had in this quiet and peaceful taverna in a small town in the foothills. Cash only.
The best seafood I have had in Cyprus, you absolutely have to book (+357 25 376082). If there is no catch that day, they are closed. There isn't a menu - they show you what they have in the kitchen. It's nestled behind the main castle square. The best parking is in the Marina carpark - one of the few covered carparks in Limassol, so it's always good to go there to get back to a coolish car!
Zygi is a small fishing town and this is a relatively upmarket restaurant there (I've seen the Cyprus and Maltese Central Bank heads eating there). It's great for lunch and you can sit under cover on the harbourside. The best parking is in a slightly scrublandish open space here.
Omodos is a beautiful mountain town with a monastery. It's a bit touristy - lots of lace shops - but it's in the mountains so a bit cooler should you want some respite. There are a number of restaurants and tavernas to grab some lunch, but don't expect anything special on the main drag!
Tsiakkas Winery is a beautifully located winery in the mountains. You can taste their wines and enjoy the views across the vineyards. I think you can get a tour of the winery too, if you want - you might have to prebook for that! Make sure you try the local grapes, even when buying wines in restaurants or for yourself. Xinisteri is always a good bet grape-wise. It's light and perfect for those hot Cyprus temperatures.
Worth a visit and a wander round. Templar history etc, with great views from the top.
After visiting Kolossi, head through the Salt Lakes (and UK Military base!) to Lady's Mile Beach. If you drive from Kolossi you will go past the Cat Nunnery, where - if you have appropriate clothing - they will let you in to meet the cats. Shortly after the Nunnery the road turns into a dust track and opens out on to the beach. You can drive across the beach to the parking and there is a beachside taverna / bar there.
Ancient Kourion is the most extensive roman ruins in Cyprus, and has an amphitheatre. Lots to walk around and explore, remember your sun protection, you'll be here a long time and there's not much protection from the sun. Also, in Waze, make sure you choose the one named "Ancient Kourion Archeological Site" as the one that is just "Ancient Kourion" takes you down to the beach. If you go in the morning, then some of the cafes on the beach below are good for a lunch by the sea.
Well worth doing the walk across the border. Take your passports. Park here and then make your way to Ledras Street - lots of shops and Cafés in the streets around. Wel worth exploring, although there are some bits that may need explaining to kids. There are some bits that you can get close to no-mans land but security is tight there so just be aware. They REALLY don't like photos of no-mans land. You cross into the Turkish side at the end of Ledras Street.
Once you're on the other side you can have a Turkish lunch in the Han (they take Euros) and then go see the Selimiye Camii - a Christian Church that is now a Mosque. https://goo.gl/maps/fr8w1r8XiVN2. They provide skirts and scarves for men and women to cover their bare legs and arms. It's worth wandering around the Turkish side a bit to feel the difference walking across a border can make.
Pafos is worth a visit, suggested parking here. It does get full so worth getting there earlier rather than later! Walk along the harbourfront, stop for food and then keep walking around the coast line, past the medieval castle for a nice seaside walk. It's not super far, but just a good leg stretch, which you'll need after driving through Pafos.
Also the Tombs of the Kings is worth a wonder around, lots of ruins and mosaics, as well as a relatively modern lighthouse.
If you want to spend a day in the mountains (maybe when you visit Omodos), then you can drive to Kalopanayiotis, a beautiful mountainside village with a small monastery with the skull of their saint in it. There's some parking in a scrublandish bit of land here. It's a couple of hours drive through very mountainous roads at least, though. There is also a wonderful spa / hotel here if you fancy a day away from madness, which also has a really nice restaurant with a view across the valley for lunch.
On the drive I often stop off at The Mountain Rose Hotel, Pedoulas for a coffee break. You can usually park on the RHS of the road as you come up to the hotel.
Near Aphrodite Hills is Aphrodite's Rock, which is where Aphrodite is supposed to have come out of her shell. Sunsets can be beautiful here.