Hi, Kristy, if it's at all possible for you to take 3 weeks, that would be ideal. You could spend 10 nights in Italy, and 4 nights each in London and Paris. That gives you 2-3 extra days to travel between the three. If you don't have 3 weeks, maybe shoot for 18 days?
Either way, I would look at flying into Rome; spending what time you have in italy there and where ever else you choose, then travel to Paris (flying there is your best option unless you want to spend 11-12 hours on a train), spend your time there, then take the Eurostar to London and finish your trip there, flying home from London.
Or reverse that. I don't think, weatherwise, it really matters where you are over a 2-3 week period. In a way, it might be better to start in London so you don't have a language barrier right off the bat, and London is really easy to make your way through. It's also very expensive as others have noted, but there are cheaper places to stay. I would recommend Premier Inn, which is a UK-based chain of hotels and they are located all over London. They are clean, comfortable and relatively inexpensive. Keep in mind that the earlier you book, the cheaper the rates will be.
Don't be afraid to come back and ask questions. I guarantee that trying to go through all these responses (including mine) will be overwhelming, so take it easy and take one step at a time. But do think about your budget. Figure out how much you want to spend overall, and how many places you can conceivably go to in the time you have. As I mentioned getting from Italy to Paris will take some time unless you fly, but flying can be expensive.
One good way to check flights is through Google, which has a site dedicated to looking for flights and giving you prices. You don't book them on Google; it's just for looking. https://www.google.com/travel/flights You can check foreign flights there, too. I just looked at Venice flights to Paris, and they range from $500 to $800+, so they can be expensive. Trains would be cheaper, but again, you're looking at an 11 to 12 hour train ride. You might consider leaving Italy for a later trip, and maybe choosing something closer like Amsterdam, or Munich and so on.
Just keep in mind it's your trip so you can to decide what you want to do.