Just know there are two main routes to learn piano, or even music:
The classical route (the traditional route). Where you go to a teacher, they stack book upon book upon book of sheet music, and 3-4 years later you're a proficient music sheet reader. after 4 years (or even 10 years) you still know nothing about chords or how to use scales for improvisation.
The 'play by ear' route. Where you don't know how to read music at all, but music theory (scales, chords) and listening to music is all that guides you and you can pick out anything from the radio. Also, this route is good for improvisation. search for 'Karen Ramirez play by ear' on youtube for this, she's the best teacher on playing by ear that I found and its all free.
When I was 13 I went about #1. I went to a teacher who'd sit me down with those painful books. I did so for 4 years, hated every minute of it. Years later I tried #2, and that's where I found my love of music again.
You mentioned you want to play that classical piece, and if so, you may need to go the traditional route. However, if you want to play anything but classical (ie, everything you hear on the radio, traditional songs, christmas songs, pop songs, create your own music, improvise, etc) then I strongly urge you go to with #2. You can get pretty damn good within a year. After a year of my going with #2 I now know about 50-60 pop songs / traditional songs in multiple keys, and I can pick up a new song in about a hour without sheet music at all. Whereas back in the day I used to know like 4 intricate classical songs and I'd spend weeks/months practicing them.
I know people that went the traditional route and 10 years later can't even play a simple song like amazing grace without having the sheet music for it, I find that really sad.