Well the first World of Warcraft version can take you from level 1-60, there are no requirements for you to completely complete the game to exceed to the next expansion.
The first expansion is The Burning Crusade, one you levelled to 58, you can proceed to The Burning Crusade content, this is an entire new zone called Outland, you start questing at Helfire Peninsula and you will gradualyl exceed to higher level zones, the majority of zones should carry you until level 70.
The second expansion which is Wrath of the Lich King, you are able to enter Northrend at level 68, again its a complete new zone and you will gradually level from questing threw various zones until you reach level 80.
The third expansion which is Cataclysm involves you questing from level 80 to level 85, there are like 5 new zones you will unlock when you get this expansion, aswell as two new races. Once you hit 85 you will get a quest to take you to Pandaria.
Mists of Pandaria is the latest expansion, you level from 85-90 by questing through each zones.
Overall you don't need to complete things like raids to access the new expansion, all you have to do is level.
Levelling is really quick and I guarantee you will enjoy it seeing its your first time playing the game.
I highly suggest that you purchase the World Of Warcraft Battle Chest. This comes with the first three World of Warcraft games.