Like Kermit said, rail sound like the way to do. Once you see the schedules and that sort of thing, your itinerary might look completely different. That's a lot of travel for 3 weeks, for sure, but you'll get to see plenty of stuff.
Amsterdam is a beautiful place, and despite its reputation as the debauchery capital of the world, it is safe, clean, pretty, etc. The same may not be said about some of their hookers.
My wife heard stories about pickpockets, drugs, etc. and made me buy a money belt because her friend made her think that we were going to be the victims of something. Not sure what, but something. The city was not like that at all, and even the red light district didn't feel unsafe. Thankfully the money belt stayed in the suitcase and remains unused to this day.
Rome is a hot (although might not be too bad in the fall - I went in July), busy, crazy place. The tradeoff is, obviously, the Roman sites and all of the history. There were just so many damn crowds at all of the obligatory tourist spots, it made it less enjoyable. Of course, we were only contributing to the problem. Places like the Colosseum are neat and you'll regret not seeing it, but I don't know that I'd go back again on my next trip. Been there, done that.
One of the most refreshing things we did in Rome was go across the river into Trastevere where the pace was a little more laid back and there were some nice restaurants and patios.
There is so much to see in Rome though... so if you're into historical sites, you'll be in heaven. But it requires a lot of metro-ing, walking, etc. and may not be the most relaxing part of your trip. Regardless of how much you're able to see, you'll always feel like you could have and should have seen more.