My first trip to Europe was Paris and London for 11 days. (We spent most of it in Paris.) So, that is one option, as they are not too far apart. You can fly or take the Eurostar. In that time we saw the Catacombs, the Eiffel tower, the Louvre, the Tuileries, the Orsay, the Champs Elysees, the Arc de Triomphe, the Place de la Concorde, the outside of Notre Dame (sob--our biggest regret is missing the interior, even though we were staying just a couple of blocks away), Versailles, the Pompidou, Les Halles, Sacre Coeur, Montemarte, the Dali museum, and Disneyland Paris. We also wandered around both the left and right banks--saw the outside of the Sorbonne, walked around the Marais, saw where Victor Hugo lived, etc. We would have gone to the Pere Lachaise cemetary, too, but a huge storm came up when we were on our way there, so we had to skip it. That was in 8 days. In 2 1/2 days in London, we saw Trafalgar Square, Picadilly Circus, Oxford Circus, Carnaby Street (shopped there), Buckingham Palace (outside), Big Ben, Westminster Abbey (toured it), the Tower of London (toured it), the Tower Bridge, London Bridge, the Wren monument to the Great Fire, (climbed it) the London Eye (rode it), Hyde Park (including the statue of Peter Pan and the Victoria and Alberta monument), Kensington Palace (toured it) and gardens, the Abbey Road crosswalk, and we attended a show (Billy Elliot). We also did some wandering around and, believe it or not, did not feel rushed.
So, I think you can see a lot and really enjoy a trip to Paris and London in 10 days.
If you choose Italy, here are my suggestions, keeping in mind that, of course, you can see only the highlights in the amount of time you have.
Venice: 2 days. St. Mark's Basilica and Square, Doge's Palace. Ride down the Grand Canal on a vaporetto. Rialto Bridge. Wander around. If you have time, take a vaporetto to Burano and wander around there. (You COULD take a day from here and add it to Rome, depending on your interests. You would skip Burano, in that case.)
Florence: 2 days. (I found 2 days to be plenty for Florence. We had 4 1/2 days and ended up taking day trips on 2 of those.) Duomo climb, Uffizi Gallery, Accademia Gallery, Boboli Gardens. Ponte Vecchio (can't miss it, really). Wander around. If you have time, there are other parts of the Duomo complex to see.
Rome: 4 days. Colosseum, Forum and Palatine Hill, Vatican museums and St. Peter's Basilica and Square, Pantheon, Capitoline Museums. Wander around. In the process, it is easy to see the Spanish Steps, the Trevi Fountain, and Piazza Navona. If you have time, consider a day trip to Ostia Antica, Hadrian's Villa, or Orvieto. Another option, if you have time, is the Borghese Gallery and gardens.
It would be a busy trip, but entirely do-able, depending on your energy levels.
You'll find that most people here will suggest a slower travel experience, and I agree that that is the ideal, but that doesn't work for everyone's circumstances.
Cheers! Let us know how it goes, whatever you decide.