The best way to begin scripting is to pick up a game engine and learn one. There are a number of free engines to choose from:
Torque Game Engine - C#
Game Maker - Game Maker Language
Flash - Actionscript
To show that you have a good understanding of scripting use it to create fun (and if you can manage it) complex gameplay within your level. Try and build upon your knowledge of scripting over every project so that you stretch yourself further each time.
Although there are a lot to choose to from, make sure you gain a full understanding of one or two scripting languages instead of gaining a basic understanding of all. Specialists are more sort after. In my opinion, with the direction that both Unreal Engine 3 and Unity 3 are taking being able to distribute to multiple platforms as well as the fact that more game companies are utilising Unreal Engine 3 and Unity technology presently I think that these are good engines to start with.
If you know of any other specialisms with the field of level design please feel free to share your knowledge by commenting.