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I was advised to avoid Hanon, actually. I only had one sit-down 'lesson' with a professional pianist, but his advice was to focus on scales and fingering and go from there. Oscar Peterson I ain't, so I can't offer anything specific. I've only been playing for about 8 months myself, after all.
Also, I am approaching piano as a studied musician. I have heaps of experience in the musical realm, and needed only to figure how to apply what I know to a keyboard instead of a fretboard. Thus, my approach might be different. If you're starting from scratch (which I now believe you are) then it's going to take a different perspective than the one required by me as a musician who understands theory and needs only to work on the practical aspect of it.
You will need to know various chords. Much of the pop material is arranged with a melody in the right hand and a chord progression for the left. You'll need to know what a CMaj7 looks like so that you can play it. Learning scales will help you to learn chords.
Studying theory would probably help. If you understand how to construct chords then you don't have to memorize which notes go where. Knowing the scales and how the chords come out of them will allow you to build what you need as you go.
roachboy's advice is highly relevant to any music, and is just as applicable to piano as it is anywhere else I imagine. At least some of your time should be spent simply exploring the instrument. Find out what new and exciting (and possibly disgusting) sounds you can make. Change things up a bit. Creativity is the heart and soul of music. Learning things by rote will make you really good at a very specific subset of techniques, but does nothing to develop range, flexibility or individual style.
Speaking as a guitarist, I know that the bulk of guitar in the bulk of styles is individualistic. You do what works for you. I see no reason why the same principle shouldn't apply to a piano.
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I wake up in the morning more tired than before I slept
I get through cryin' and I'm sadder than before I wept
I get through thinkin' now, and the thoughts have left my head
I get through speakin' and I can't remember, not a word that I said
- Ben Harper, Show Me A Little Shame
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