I would start in Rome as well. One other reason for doing so is that there are more cheap flights from Rome to either Paris or London (check prices). Also airfare to Rome as been priced quite competitively lately.
You should also consider scheduling your trip leaving home a Thursday and flying back a Sunday, instead of leaving a Friday and returning a Monday. Generally, you are likely to get a cheaper airfare using those travel days.
Monday is also a day when many museums close, so that makes it an ideal travel day. With only 2 full days in each city, I would skip doing any daytrips at all (except for maybe Versailles in Paris).
For sure, do the Hop on Hop off bus in Paris on arrival - that one is actually quite good (don't bother in Rome or London, IMO, as I feel that you'd be wasting too much of your short time there). Be aware of where you'll encounter the most line-ups and pre-book, if you can: Colosseum, Vatican, Eiffel Tower, Louvre, Tower of London (even in early May).
Leave home Thursday
Friday to Monday => Rome
- start your first day by walking around Rome, meandering between the Spanish Steps and Piazza Navona (Trevi, Pantheon), maybe do a 'Rome by Night' bus tour
- pre-book Vatican Sat morning, walk Trastevere & the ghetto, plan a 'splurge' dinner
- pre-book Colosseum for Sunday, visit Forum
Monday to Thursday => Paris
- fly midday (noon-1pm) on Monday FCO to Orly or CDG (not Beauvais!)
- pre-book Eiffel Tower visit for Monday night, climb Arc de Triomphe
- Versailles (morning) and Montmartre (late aft & dinner) on Tuesday
- Louvre on Wednesday, plus Notre Dame, Sainte Chapelle (open late Wed), boat on the Seine
- fit in one more scheduled visit on Thursday morning (ex: Orsay museum or Père Lachaise cemetery)
Thursday to Sunday => London
- train to London as late in the day as you can manage on Thursday, hit a pub
- go to one of the Saturday markets in the morning, a museum in the afternoon, and a pub crawl in the evening
- Tower of London early in the morning on Sunday, bus to Greenwich, boat back down the Thames to Westminster, and a show in the evening
Sunday you fly home
This would give you a fair amount of time to see the highlights in each of the 3 cities, and it's very doable as you can see from my suggestions... go for it!
As Ken mentioned above, I also recommended taking with you the Rick Steves Pocket Rome / Paris / London / guidebooks. :-)