I appreciate all the comments made by the good folks who want to help you in your new path to music.
That being said, I want to trump them all. I am a very accomplished pianist, having, AHEM, fifty years of hard practice and god-given talent and love for my favorite hobby. Many here have heard my music. Most like it. Wannabes critique it. But wannabes don't offer anything other than their words and criticism. Those who cannot do it, bitch about those who can do it.
I play Mozart (the Dm concerto and a handfull of Sonatas and his Cm Fantasia), Beethoven (Sonatas [Pathetique, Appassionata, Waldstein, Moonlight] and the #3 Concerto, the Choral Fantasia and yeah, Fur Elise, and many more), Rachmaninoff (Preludes and the Fantasy on a Theme by Paganinni), Brahms (Rhapsodies and more), Liszt (Liebestraum, unSuspiro Etude in D flat major), Schumann (Arabesque, Traumerei), Chopin (several Etudes including the Cm Revolutionary, the Fantasy Impromptu, Waltzes and Preludes), Bach (three Preludes and Fugues [Cm, Gm, B flat M], all the trivial 2 and 3 part Inventions, and the French Suites, Partitas and most of the Chorales), Ravel (Pavanne), Debussy (many things including Claire de Lune, The Engulfed Cathedral and most of the Children's Corner), over a dozen of Dave Brubeck's stuff, and lots of other popular fun stuff like the Peanuts Linus and Lucy Song to name just a smidgeon of what I have learned and played over the years.
Here is my humble advice for an adult who wants to learn piano: YOU HAVE A MAJOR ADVANTANGE OVER THE KIDDIES! You have the motivation and you don't have the distractions that kids who are forced to learn an instrument when they don't care to do so.
Ask your teacher if he/she teaches solfege. If not, find another teacher. Solfege is nothing more than do, re, mi, fa, sol, etc. but it teaches you to sing the notes of the scale. Even if you cannot sing on pitch, you will have a great head start on those who do not learn it. Trust me on this. If you can even come close to singing something you can come much closer to playing it. I can not sing 3 notes in pitch to save my life, but after trying to sing them, I can hear what they should be in my head, and I can play them.
Another piece of advice: if your teacher uses progressive music books like the kind that starts you with music book "A' and you graduate to book "B" and so forth, the teacher needs to be fired. I went through that garbage and didn't learn a thing about music. I wasted three years of my life on that crap and didn't learn a thing. Same thing is true with scales and the "Hannon" books. Waste of time. Also, don't bother with the "Czerny Etudes" . Czerny was a brilliant pianist who premiered several of Beethovan's Sonatas and Concertos when Beethoven was deaf. He was a brilliant pianist, but he was a pathetic composer. Better to learn Bach's 2 or 3 part Inventions which happen to be REAL music and which you can play for your own pleasure and for the pleasure of others or as you progresss, Chopin's Etudes (exercises) which are also REAL music than to be bothered with meaningless Hannon scales and Czerny exercises. Trust me. I know what I am talking about. There is great music that will teach you technique and will still be great music that you can play and people will enjoy. You cannot play Hannon or Czerny to an audience. That crap is not music.
I've rambled long enough. Actually, I have much more to say on the subject, but I've said enough for now. Just to give you an idea about the dedication required, I will tell you how long I have worked on the Rachmaninoff "Rhapsody on a Theme by Paganini": over two years! I'm 90% done to my satisfaction. More talented people could have tackled it in say, 5 months, and I'm talented. There you go. Making good music is hard work.
Nothing worthwhile is easy, but something worthwhile is worth all the time it takes to achieve it.
If you would like to hear some of what I can play, PM me and I'll be happy to send you a CD containing a sampling of my work.
Cheers,
Farkel