Avoid Heathrow at all costs. It's a fricking nightmare - possibly the worst airport I've used. And after experiencing over 150, that's saying something. Gatwick is much easier to deal with, while Luton is just crap.
Barcelona is a must IMHO. Prague seems to have lost a bit of it's
je ne sais qua, but will always be worth a snoop around.
I'm not much of a fan of London or Paris really, and can't really remember Rome (student, drinking, you know the story).
I'd really ask what kind of thing are you interested in seeing. Europe pretty much has it all, so tell us if you're more in to seeing hills and crinkly bits, quaint tea rooms, sheep, vine yards, etc etc.
On a week long whirlwind tour, I'd limit myself to two. These are big places you're talking about, with long travel times between most (e.g. Barcelona is a good 4hrs from London, and that's without all the check in and waiting). If you must fit in three, look around for the quickest ways between A and B -
The Chunnel is a great way to get from London to Paris without much hassle, and you get to see some countryside instead of just sky. It usually works out faster than flying, too. Plus it's fun to ride a train that does close on 200mph.
Taking in the 'other' sights has always been a priority for me when travelling, so get some books, get online, and read up on the less well known, or well travelled bits. Barcelona is fantastic, but there are even better bits hidden away in the countryside round about.
In the end, only you can really decide where to go and what to see, so how about just asking if people can post links to information about their favourite European destinations, and pick the ones that look most exciting?
