Once i buy my first bitcoin, what exactly do i do with it?
Well you can spend it to whatever you want, or you can just store it 'til the price increase 
Do i leave it in my coinbase account? Is it safe there?
Or do i transfer it to my desktop computer?
Yeah, I think you have to send them to your desktop computer wallet. It's just like when you withdraw cash from a bank to your wallet.Or do i transfer it to my desktop computer?
And if you want to be more safer, you can try Armory. Or if you do not plan to send the Bitcoin often, you can try Paper Wallet.