ok go to wow-europe.com
access the account management, here's the fast link(
https://eu.blizzard.com/login/login.xml?referer=https%3A%2F%2Fwww.wow-europe.com%2Faccount%2F&loginType=wow)
enter your account name+password :p :p and your account management page will be displayed scroll down the page and you'll be to see "Purchase classic world of warcraft", press that and you'll be lead step by step until you pay(15€) and your account is upgraded, when you do this electronic purchase, your account is automatically upgraded, you don't have to enter a cd key..
Now you'll be to see that your account type is world of warcraft classic, you'll be able play and travel around in 2 continents, eastern kingdoms and kalimdor, you'll be able to reach level 60
So now(after buying classic WoW)you can buy the purchase the burning crusade(15€) using the same way, find the link in account management, you'll be able to visit and new continent outland and a couple of previously locked areas in old continents.
After that you'll be able to purchase Wrath of the lich king using the same way, it is the most costy(40€), you'll be able to visit a new continent(northerend), level to 80, and play the big events(raids) that everyone is doing/attempting to do.
so the total costs will be 15+15+40=70€ + monthly subscription of 15€ to play every month
Note that you can purchase world of warcraft plus its expansions directly one after the other or just purchase classic wow, play until you are level 60 mess around, then purchase the burning crusade do the same, then finally get wrath of the lich king. Note that you can't get them in any other order :p...
The only real incentive to get wrath of the lich king are the raids(big events), raids in classic wow and the burning crusade are not played anymore or too few play then so you won't be able to find a group for there, but raids are still too few in wrath of the lich king, so play the game at your own pace(imo take it slowly, savour every moment of it, being a newb is not bad) and pay at your own pace...
I should mention that when you buy burning crusade you'll have to download burning crusade expansion and when you purchase wrath of the lich king you'll have to download the expansion too, that's why lebscream's one is 13gigs, but if you are not going to purchase them directly, you do not need to have them on your pc but imo start trying to grab them from somewhere, but be careful when you purchase an expansion you are required to have it, you'll need to wait until it downloads and installs to play or of course already have it instaleed on your pc..
I hope that was helpful and not just a block of text :p