I’ll also heartily recommend Rick Steve’s “Europe Through the Back Door” book. There are some great tips there in there.
My wife and I took two trips similar to what you’re planning (the home page linked through my profile has pictures if you are interested in seeing us at some of the places you mentioned).
Some advice:
London: never been there, but its likely going to be the most expensive part of the trip b/c of the exchange rates. You may want to consider France alone and get to other places, such as Colmar/Alsace (very germanic) and Chamonix (especially if you ski, or want to see some incredible mountains).
Plan your itinerary in advance to maximize your time available to see stuff. You’ll be using the train systems in either France or Italy - both have schedules posted in English on their websites.
Both France or Italy - buy a museam pass the minute you get there. Its more economical than paying to get into each museam individually, and will automatically get you to the front of the line in most places. Its a great time saver.
Consider also buying a Eurorail train pass - its the only way to get from place to place, and again more economical than buying individual tickets if you’ll be seeing a lot of either France or Italy. Some passes come with a day or two of car rental - this is a necessity if you hope to see a lot in the South of France or Tuscany.
Fly “open-jaw”: in one city, out another. It should be the same cost, and will prevent you from wasting a day backtracking to your point of origin.
www.venere.com is a great website to scope out hotels/places to stay anywhere in Europe. You can search by local, rating, cost, etc.
If you really want to maximize your time, plan train rides overnight and sleep in a cuchette (typically, you’ll be splitting a car with fold out beds with 2-4 other people). I personally didn’t enjoy the experience b/c I am a very light sleeper, but it was doable. Be sure to bring luggage locks to protect your stuff.
Some opinions:
Rome is great for the history, but we tired of the pace/noise. You can see most of the good attractions in 3-4 days.
The Almafi coast (south of Rome, near Naples) is very nice. Sorrento, Positano and Capri are beautiful seaside/island towns. You pass through Pompeii to get there, which was also cool. Stay away from Naples - its a wreck.
The Cinque Terre is also beautiful. Its served by trains, and you can hike all 5 towns within a day. Its incredibly beatiful (often referred to as the “poor mans Riviera”. While you’re near there, consider the Lake Como/Bellagio area. Also beautiful.
Florence and Venice were both great with plenty to do, but we enjoyed Tuscany the most. However, you’ll need a car to get around there. Its really not too bad to drive in country. Ditto for S. France - you’ll want a car to really see it, and the country roads are fine for novice drivers (don’t take it personally if you get honked at 
You can literally spend a month in Paris and not see it all, so prioritize.
To be honest, I enjoyed our France trip a little more than Italy, but both were great. France seemed a little more “tourist accessible” to me, but neither country was a hassle.
I did feel like we had to be a little more cautious in Italy from scams and pickpockets. (if someone flashes a badge and demands to see your ID, just walk away) - all police there must be in uniform when on duty. This is a common scam to get you to take out your wallet (speaking of which, get a money belt that cinches inside your pants around your waist - its the safest way to carry cash/ids).