Piano Playing Tips

I know there are other Piano threads, but this is for me.

I want to learn how to play the piano properly, and I need some tips.

What I already know:

- I'm a Gr. 7 (ABRSM) Trumpet player, so music is no problem whatsoever. Treble Clef is as easy as breathing, and although I'm slow with Bass Clef, I can read it.
- We've got a piano, and I've played with it for a couple of years, so I know where the notes are.

Erm... that about covers it, really - I'm already a musician, so I know the clefs, notes, and where things are on the piano.

I can also play things with my right hand (more than one note at a time is a bit taxing, but I can do it after a few minutes practising), though my left hand isn't fantastic. Two hands together is something I just can't do very well at all.

Edit:

I've also got longish fingers, and can touch type, if that helps at all?

Currently there are three pieces I want to be able to play: To Zanarkand from FFX, This is Halloween from Nightmare Before Christmas, and Casper's Lullaby/One Last Wish from the 1995 Casper film.

Short of sitting at the piano for hours and playing the pieces until I can do them well, does anyone have any tips? I'd particularly like to be able to play a new piece by sight in the future, and I don't think that forcing myself to learn three pieces would allow me to do that.

My choir training has allowed me to be able to pluck out notes from sheet music on a piano, but my fingers are so short that I can't really learn to play effectively, chords and such, as much as I would love to. I used to be able to play the main melody of Eyes on Me - FFVIII, and I probably could if I sat down and tried to pluck out the notes again. I can play the main melody of Rainy Day Man from Sailor Moon still, though...