The "best time to go" really depends on your interests and your reactions to things, both positive and negative. For instance, I really don't like dealing with rain and chilly weather when I travel, so I avoid those months UNLESS I am going to a destination where the main atttraction for me is museums, and then I don't care. In fact, I often pick lousy weather months to go to someplace like London or Amsterdam, because there are much fewer tourists in the museums.
But also be aware that it can be 100F (40+C) in Florence in the summer (and most of central Italy), plus loads of crowds. It can also snow quite a bit in Florence between the middle of December and the middle of February.
There are definitely some types of travelers who truly don't care what the weather is so long as they are in Italy (or Paris, etc), and who never mind sharing space with other tourists. The more the merrier! But if you have negative feelings about extremes of weather, or if you really don't want to be in peak tourist season (the crowds in Florence can be intense, every day, between June 15 and Sept 15), then plan accordingly.
As to where to go for your other 2 spots in Europe, if you don't want to rent a car, then look for "bases" with excellent train connections to the types of experiences you will want to have over 30 days time. For instance, someplace like Antwerp is actually pretty astounding for beer, food, art, history, nightlife, music, shopping, terrific train trips to Amsterdam or Paris (you can go more than once to each city) -- and yet, it is off the radar for mass tourism, so you can have a pretty fascinating experience of Europe, unwarped by travel commercialism.
On the other hand, 30 days in Paris or London is non-stop excitement right outside your doorstep every day, but also with the possibility of some wonderful day trips by train to Antwerp or a few nights in Amsterdam or parts of Germany.
Something else the weigh into your thinking is how much you want to see the most famous sights of European history and how much you would like to be spending time in places that are NOTHING LIKE wherever you come from. The most famous sights of Europe make it easy for visitors, by necessity, and it helps make sightseeing relaxing and rewarding. Some parts of Europe are very modernized, and you'll find many of the same chains found in America and the UK serving coffee, pizzza, etc. But there are still many parts of Europe that dance to their own tune -- surprisingly, a great deal of Lisbon remains that way.
Lastly -- then I will stop! -- the European continent has some extraordinary scenery -- and if you like walks and rambles, or swims and boat rides, you can enjoy some blissful times with wonderful food and drink, usually plenty of history under your feet, and not necessarily all that far from some amazing art.