classical piano recommendations?

Similar to the first movement:
Debussy Clair de lune (obviously)
Grainger-Fauré Clair de lune
Most of the John Field nocturnes
Chopin: 9/1, 9/2, 10/3, 15/3, 34/2, 37/1, 37/2, 48/2, 69/2, Berceuse, etc. (for the ones with numbers, search for 'Chopin op. 9 no. 1', etc.)
Fauré: Bacarolle 1, Nocturne 3, 4
Liszt: Transcendental Étude No. 3

Similar to the second:
Scherzo from the Moonlight's brother, Beethoven 27/1
A lot of Chopin's mazurkas,

Similar to the third:
Liszt Transcendental Etude 11
Chopin Fantaisie-Impromptu

Mood of whole piece:
Chopin: 27/1, 27/2, 48/1, Ballade 3
Fauré: Nocturnes 1, 5

Ignore people suggesting Satie and Debussy (Claire de Lune) if you are looking for heavy notes! They are very light textured!

Go for Chopin piano concertos and sonatas, anything by Rachmaninov, Beethoven piano concertos and sontatas (the later ones tend to be heavier).

https://www.youtube.com/watch?v=vUWP3KPLOao

Rachmaninoff - piano concerto 2

Crazy piece but you may or may not want the orchestra accompany the piano.

Some parts are pretty up beat but it is an interesting song.

In keeping with your liking the Moonlight Sonata, try Debussy Claire de Lune.

I personally love Erik Satie's stuff
Some pieces to check out are:

http://m.youtube.com/watch?v=iHEpuj96bCg&list=PL4ccjKA-sPuBhHQoUXABqealW_tuv1GTI

http://m.youtube.com/watch?list=PL4ccjKA-sPuBhHQoUXABqealW_tuv1GTI&v=i8356FxUT20

http://m.youtube.com/watch?list=PL4ccjKA-sPuBhHQoUXABqealW_tuv1GTI&v=S-Xm7s9eGxU