Just to put it in perspective, this is the time you have for sightseeing:
London - 2 days, when you are still jetlagged and catching up on sleep
Paris - 2 days
Amsterdam - 1 day
Berlin - 1 day
Venice - 1 day
Rome 2 days
Barcelona 2 days
If you want to take a trip at a pace like this, you should book an RS tour. You'll have door-to-door transportation, a guide to skip the lines and show you the main sights. Otherwise, you are going to spend way too much time getting oriented, getting to/from airports and train stations, and missing out on the best experiences when you get lost or, worse, your group wastes a lot of time deciding where to go and what to do. On a tour, there's pressure to be on time and keep to the schedule. 4 people on their own . . . someone's going to sleep in or linger over breakfast. Someone's going to want to stop for a break or a meal. And unless you are all easy-going and flexible, three weeks of whirlwind can be a recipe for fraught nerves, frayed tempers and worse.