Pretty much all my life I've been able to play music by ear, and have played piano just as long, but I only took a few lessons on and off, here and there, and I've been complacent to tinker around with short, easy little songs from popular tunes, not using both hands most of the time... I'm sure a lot of 5 year old music students are better than me at this point, and I'm 21. I look back and think, "Man, things could/should have been so different... I should be playing advanced classical/jazz upside down with my eyes closed by now".

So, how do I catch up? I bought a book that teaches how to read music, as well as piano techniques and even a bit about composition, improvisation and music theory... is it imperative for me to take formal lessons?

I'd love to get as good as Billy Joel...