I'm looking for a warm, not too expensive place in April, not more than 3 hours from the UK, and not Spain. We have a 3 year old and one year old. I saw a beach in a guide book, called Mondello outside of Palermo in Sicily, and wondered how kid friendly it is? Also accommodation that would be very quiet so we could all get a good, early sleep at night. Would stay in Palermo if it was cheaper.

I don't know about Mondello but you might want to have a look at the vacation rentals available through homeaway.com (by owner) here: http://www.homeaway.com/vacation-rentals/italy/sicily/r283
There are a huge number of listings for Palermo and you could probably find something quite reasonable for that time of year. Especially versus a hotel.
If you haven't done a vacation rental before, look for reviews and a property that's been listed for a while on the site plus a calendar with at least some dates taken.
We did all our vacation rentals on the Europe portion of our year-long RTW through this website.