You will write down your flight time after each lesson and often the instructor will watch you completing your logbook, certainly where I am anyway. My training record also shows the flight time so I can double check. Some lessons I also have to get the instructor to sign my logbook as evidence of the activity completed. I’d get a logbook as soon as possible.
There will definitely be a record of your trial flight in the aircraft tech log - they might have shown you or you might be shown on your next lesson as eventually you’ll be completing it yourself.
There’s a Pooleys starter kit with everything you need excluding headset. Some of it you won’t need straight away. Or you can buy it all separately. I had a mix of Pooleys and AFE books/equipment for the PPL.
Read up on headsets, David Clarke’s are about £300 or Bose up at about £800+. I went for David Clarke’s, not had any issues. I have heard students mentioning the Bose A20s cause a lot of compass deviation in the 152.