I paid a local bike mechanic to go over my new bike with me, get it set up correctly and instruct me in the basics. This covered handlebar/stem adjustments, wheel removal/installation, tyre and tube changes, brake adjustments, brake pad alignment, pedal fitting, seatpost and saddle setup, mudguard/rack fitting and setting up a wired computer. This cost me £40 and took about 2 hours, he also recommended some basic tools for me.
Since then I've expanded slightly to include brake and gear cable installation and adjustment. Indexing the gears and replacing a chain, splitting and rejoining a chain as well. As I go along I've been investing in decent tools for each job. My next aim is around chainwhips for cassette removal and tools for chainsets and bottom brackets. Headset installation I still leave to the LBS. I'm leaving wheel truing/building to the end - it scares me
I think just play it by ear and go as far as your interest takes you.