So, to compare: I've got Abitur (= German thing, similar to A-Levels), and an apprenticeship as a machinist. Then I went to university to study robotics.
I learned programming, game programming and specifically Game Boy programming (the old B+W Game Boy at the time) on my own time, which earned me an offer to go to the UK to make Game Boy Colour games for a well known big developer at the time.
Since then I worked on Game Boy Advance, Xbox, Xbox 360, PS3 and Wii, mostly picking up new things on the go.
We used to recruit new designers from the QA/Testing team, but I'd advice anyone against planning that as your career path: For every tester going into design there are many who spend years testing games at low pay only to be laid off the moment a project is finished.
If you're interested in art it is possibly safest to go through a traditional course to give you "proper" training, so you can branch into TV, movie, advertising or games work. You can pick up many of the game specific techniques as you go, there are lots of forums like polycount to exchange knowledge. It is hard to actually teach things like that in a course, because many things have changed significantly over the last decade. Just compare playstation and N64 games, where you didn't have the fill or transform rates for complex scenes, to the latest PS3 and X360 games where you not only have highly detailed models, but cloth simulations and complex lighting and shading to consider.
For design it is a bit harder. Nobody needs just someone with an idea: Everyone I work with has many, many more ideas than we can ever use. Nowadays you need basic art skills to rough out ideas and levels (like the
orange-maps in HL2), and a bit of programming to script game events and control AI characters. You need to be able to work in a team, and most importantly be able to design within the limitations of the budget and engine.
Our industry is still to young too really formally train game design (as opposed to thousands of years where people drew or painted things), but just playing isn't a good training. You've got to understand why and how a game works, and even understand and be able to design games you don't like.
There are many free or cheap ways to get started. I think Autodesk has 30 day trial programs for Max and Maya, there is
Blender and
GIMP for graphics.
You can play around with
Unity,
Unreal,
Game Maker,
Multimedia Fusion or
pygame. And AFAIK if you own one of the Source games (HL2, Counter Strike, Team Fortress...) you can download the Source SDK from steam. If you want to make an RPG then
RPG Maker XP is a good starting point.
Before you start with your epic game it is a good idea to start just with a single level for another game. If you fail there you haven't spent a fortune you'll never see again.