The above is incompatible with the itinerary you give below: That's at least 10 cities in 3-4 weeks! Very much an "if it's Tuesday this must be Belgium" type of tour: you'll be spending half your time making travel arrangements, packing, unpacking, checking in and out of hotels, and going to and from airports and train stations, and everything you see or do will go by in a blur.

People will give you various advice, and not everyone will agree with mine, but having made a lot of such trips, here is what would would work better for me:

• Stay at least three nights in any one city, except maybe for the first and last nights of the trip. Yes this means seeing fewer places.
• Unless there's something specific there that you're interested in, skip Hamburg. Stay the first night at an airport hotel (these are often relatively cheap) to get over jet lag, then continue on. Berlin is very worth while. If you want another German city, try Munich.
• Likewise I'd skip Milan unless you have some special reason for going there.
• Of the other cities on your list I'd give top priority to Paris. I'd also recommend Vienna. Venice is wonderful but it's so expensive and tourist crowded that I'd skip it this trip unless it's someplace you've always dreamed of seeing.
• In Switzerland one of the best things to do is take one of the famous train rides, like the Glacier Express.
• People will give you different advice on car rental, but generally you don't need one if most of your time is going to be in major cities. They all have good transportation between them, and in the inner part of most European big cities having a car is a disadvantage (parking, traffic.)
• Three nights in Athens -- two full days -- will be just enough to see the major sites and museums (including maybe a side trip.) Mid-price hotels in Athens pretty regularly cost about 150 EUROS an night. If you also want to go to Greek islands, I'd spend at least a week in Greece. From Germany (or elsewhere in northern Europe) to Greece, fly.
• For a trip like this it would make sense to investigate rail passes and air passes (the kind an alliance sells if you buy an international ticket from one member airline -- some people will argue that low cost carriers would be cheaper but the passes also work out to be fairly cheap and are more convenient so personally I prefer them.)
• Best guides I know in general are the Rough Guides (for independent economy travelers but not just backpackers) and the DK Eyewitness Guides (especially good for first time trips.)
• Despite their attractions, on this trip it might make sense logistically to omit either Britain or Greece.

There will be followups taking exception to some of the above, but as I said, it's what has worked for me based on experience.

Enjoy your trip!