I sing and play piano and take lessons on both if that helps. I am doing grade 5 piano and just completed grade 8 on singing. I can self accompany (not really well but I can do it).
Heres what I would say to someone starting out...
(1) Learn both parts separate like you would learn something hands together.
(2) If you know know the melody learn it by playing the phrase and then singing it.
(3) Once you can play the piano part really well that it is locked into your muscle memory to the extent you can play it blind folded then add in the vocal part.
Start off just with one hand with whole block triads to supply harmony for yourself and sing over it as an exercise to listen to your own voice and get used to it being accompanied with harmony.
Look at pop songs, especially ones with piano accompaniments (Evanescence - My immortal, Delta Goodrem, Missy Higgins, Britney Spears Every Time, Vanessa Calton - A Thousand Miles, Elton John / Billy Joel - Pianoman, Kelly Clarkson - Because of you, Rufus Wainwright - Hallelujah) - just a few suggestions. Listen to them and maybe learn a song or two you like. You can clearly hear the piano part with the vocalist and easily learn it.
Hopefully that helped. I know its tricky but the more you try and practice it the easier it becomes (bet you've heard it before). :;):
Music is organised sound