For a first trip to UK/England, France and Italy--with a week in each spot--I would recommend the following places (in summer):
England/UK: London (also, don't forget you'll need to recover from the 5 to 8 hours of jet lag), Warwick Castle near Coventry, town of Bath and the Cotswolds area.
France: Paris, Nice, Monaco (yes, actually a separate country, but entirely surrounded by France)
Italy: Rome, Tuscany (specifically Florence and Siena), leaning tower of Pisa, Venice, and the lakes region north of Milan (Lake Como, Bellagio, even Lake Lugano across in Switzerland).
P.S. The lakes region is a good place to cool down after visiting the normally very hot locations of Venice, Florence, Siena, and Rome.