I will travel to Europe this year. I am planning to visit Manchester Old Trafford on the first day. How can I get to Manchester from London? How long does it take, and how much does it cost (preferably by bus/train).
Tackling the London-Manchester bit, Virgin Trains runs regular services on this route. If you book in advance, tickets should be under £20 each way. There are several fast trains an hour, and it takes about 2 hours for the trip. You can get timetables and book tickets on the virgin trains website (amongst others)
One note about UK advanced train tickets - tickets generally go on sale 3 months in advance. The very cheapest tickets will often sell out that week, but you can normally get some fairly sensibly priced ones for several weeks after. (There are normally advanced tickets at a couple of different price points for most journeys). Finding out exactly when tickets are on sale to does vary in difficulty, First Great Western as an example display their booking dates on their homepage, others don't. Another snag is that tickets aren't released until engineering work is announced. This may mean that weekend tickets go on sale later than weekday ones, and some companies will have their weekend tickets available weeks before others!
Advanced purchase train tickets are almost always for a specific train, and non changeable. They are generally only available on longer journeys, but you can sometimes get a local journey at each end included (depending on the company). I think almost all the train companies offer them, but advanced tickets are generally tied to just that one company. This means that if you want to do a journey that involves two companies, you'll often need to book two advanced tickets...
Finally, if you're booking fairly last minute, expect things to be rather expensive, especially on popular routes. If you're booking on the day / almost on the day, you may be able to save a little bit by taking a slower train. Ask at the ticket office about that, and they'll tell you if it's possible. Another option that may be available is Split Tickets, where you find that splitting your long journey up into lots of local legs can be cheaper!
Oh, and if you're travelling off peak and with a child, then get a Family Railcard. There's an annual fee, but you'll generally save that on any single long journey. You can't start too early, and you must have at least one child in the group, but for family leisure journeys it can save a lot.
Not exactly an answer, but a word of caution. It is important enough that I didn't want to make it a comment.
Last year I went to Italy and got a data stick from Vodafone to use while I was there (working). Of course, I don't speak italian, so I had to rely on the customer rep to explain the options to me. I selected what seemed to best fit my situation, but I had to sign a contract to do so. I was told that I can simply cancel at the end of the trip, no charges, no problem. It turns out that 8 months later, I am still being charged by Vodafone Italy. I cancelled the credit card, but because of the way they charge, Master Card gave them the new card number. Master Card seems unwilling to deal with the situation and all I have to prove my point is what some customer rep told me 8 months ago. The charges are still in dispute.
Conclusion: don't sign any contract that you can't read and even then, you better read all the fine print.
You can get a global sim that will work throughout Europe, the prices for data are on the expensive side, you might want to read some of the relevant answers to this question - about avoiding data roaming fees.
As for trains, all British train information, including booking of tickets, can be found at The Trainline. Prices and journey times vary depending on your route, faster more direct travel is usually more expensive, and buying well in advance can save a lot of money.
Vodafone works in pretty much all european countries; however at 15Euro for 50MB for traffic (Vodafone.de prepaid roaming data rate) you are better of buying a new (local) sim card in each country.
t-mobile uk offers a cheaper rates (up to 1 Pound/500MB per day), and im pretty sure it will also work in all the other countries on your list.