Sklithri beach is peaceful. Sand is firm, some seaweed in the sea.
Troulos beach, far right side is a good morning spot. Does get busier as the day goes by. Can get a bit wavy when big boats come in. So around 10ish and again 10:15ish, then on some days 10.30ish.
Vasillias beach tends to be quiet/not as crowded. A bit pebbly as you're entering the water and in some parts flat rocks underfoot.
Note that the buses do get very busy in the evening, so depending where you're staying, if you chose to come into town at night, the water taxi would be a better option for you. The resorts out of town are much more relaxed in the evening, town can get quite stimulating with a lot of walking traffic on Pappadiamanti Street. Agree with Alexandra's comment. You'll all have a great time if you have beach mornings as most of the beaches are very quiet early on. Pools tend to be less busy, so would be greater for the latter part of the day.
There is now a little fun fair (bunjee trampoline, trampoline, mini boats, bumpy cars etc) which isn't too busy depending on the day (not a weekend activity).