If you buy one pre-assembled, it is one part.
Some good parts to get would be:
solderless breadboard
wire
LEDs
potentiometer
piezo
10k resistors
pushbuttons
I started with the blink example on the arduino tutorial page, and after you see how it works, try and editing it: maybe change the speed of blinking, adding syncopations, etc..
Then I would learn how to hook up pots and buttons. The first project I made without a tutorial was an led that could change brightness according to a potentiometer.
Also, the tutorials on the arduino website are useful.
But I would recommend actually buying a kit and assembling it yourself. I did it when I was 12 so it is not actually that hard and you learn a whole lot more. It is also slightly cheaper. (Quick disclaimer, I have been taking apart computers since I was 5, soldering since I was 7, and coding since I was 10.)
I think, thas the best way to learn arduino is the good idea. Have an Idea and go it.
Do not forget about wiring and processing - with processing and arduino you may does a lot of interesting things!