I've just had my first piano lesson as a proper "grown up". Am tackling 3 of the easier Chopin preludes to begin with and have been shown fingerings for all the major scales too. It's very exciting...

Does anyone have tips on methods of practising? e.g. some people do lots of practice with separate hands; others go for both from the beginning. Also, do people tend to think in terms of notes, fingers or harmonic progressions when playing piano?

Hands separate or together tends to depend on what I'm learning.
Some things I need to do separate hands and some things work better if done (mostly) hands together from the start.

When play piano I read patterns rather than notes names (apart form finding out which note to start on of course).
You learn to see intervals and chord shapes as well as other patterns in the music.

I parctise by playing throught the piece, and stopping every time I can't do something and working on that particular bit and then carrying on. Sometimes when I know there's a certain bit i can't do I start by practising that bit before going fom the beginning. If there are fast quaver/triplet passages I practise them dotted and different rhythms, but I'm sure your teacher would have told you that

I tend to think patterns on th piano, but if it's something really complicated like a Chopin polonaise I practise until my fingers know where to go without thinking That does take a long time for me, usually a few months...

Normally when I start a new piece, I will play one time for the right hand and one time for the left hand just to get the melody, fingering or any accidental and then I will play with both hand.

The above is true, but I'd be wary of giving the advice to a comparative beginner. It presupposes a fairly advanced knowledge of the theory of music when, after a while, you become more aware of the progression of chords etc. At the start it tends to be a case of "work-it-out-from-top-to-bottom" without necessarily having much idea of whether or why chords are arranged as they are.

LR

benjaminja is a post-Dip violinist and very capable on at least one other instrument (maybe more, I forget ), so I suspect that he won't be too bamboozled by sbhoa's advice...

and it was meant to be 'what you are working towards' sort of advice rather than how to manage at the start.

I had to relearn the piano on several occasions, normally after the times I had to re-learn to walk. I get serius migraines and they have the habit of paralysing me temporarily down one side, leaving me with little co-ordination for a while.

The discipline is normally really useful, but can be very frustrating going from grade7+ to grade 3 for a couple of weeks is maddening.