Hi Nera84,
bentivogli is correct in saying that you need a Schengen visa which will allow you to travel within the borders of 23 of the 25 EU member states.
But as you quite rightly pointed out, both the UK and Ireland are not part of the Schengen area agreement even though both countries are EU members. This is because they both used an opt-out clause in the Treaty of Amsterdam which allows countries to maintain their own border controls.
This means that you're going to have to obtain a Visa for both the UK and Ireland in addition to the Schengen Visa.
As regards travelling alone, you mentioned that you've been to a number of other Asian countries already, so I would imagine that you're an experienced traveller by now and would be able to judge people pretty well. You're not planning to visit countries like Bulgaria or Romania, so you should be OK as far as your own personal safety is concerned.
Whether you'll be able to cover them all in the 3 weeks you've planned for the trip is another matter. You've listed 16 cities which is quite a lot to see in just 21 days. If I were you, I'd miss out on Birmingham (nothing of interest there at all) and go to London. Take an Easyjet flight from there to Amsterdam (where I live) and then pick up your train again. Miss out on Hamburg which is in the North of Germany and replace it with Munich which is in the southern half and easier to reach from Amsterdam. From there, you can hop across to Paris, but skip Bordeaux (unless you've got a passion for wine). Skip Milano and go to Venice instead (you should definitely go to Venice, it's just so romantic), and from there, go to Florence (another Italian city well worth visiting but not on your itinerary) and finally to Rome. If you want to include Dublin, you could consider using that as your initial destination airport.
Forgot to include a map of the Schengen countries, so here it is: Schengen countries
Also, some Visa info which you might need: EU Visa policy
Oh, and the Youth Pass sounds like a good deal. I doubt if you would save anything by buying individual tickets.
[ Edit: Edited on 23-Apr-2009, at 09:31 by Blinq ]