With regard to supplies, buy everything when you are in the US. You would be a bit silly to buy it prior to going. The exchange rate favours doing it all when you land out there - just remember to leave space in your luggage for the return journey!
Of course, do a bit of research with regard to which headset you want to buy prior to going, etc.
It might be even better to order it up a few days before you arrive so it is waiting for you at the school upon landing.
For sectionals & an AFD, the local pilot shop on the airport. For everything else, get it ordered with an online US pilot shop with free delivery & huge savings on UK prices.
You will only need one logbook - it can all go in the same one. Get a JAR one.
Regarding the checklist, not sure how the particular school operate, but I would imagine the school will issue you with their own one. Same goes for the POH.