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.