It's not that unusual that schools don't bother to mention a logbook for the first few lessons, as it's easy enough to retrospectively enter the first few flights from school records.
If you're going professional, I'd buy something labelled as either "FAA Professional" or "EASA compliant professional". For example...
https://transair.co.uk/pilot-supplie...-book-non-easa
https://www.pooleys.com/shop/pooleys...ying-log-book/
https://www.afeonline.com/shop/afe-p...s-logbook.html
(Those three links probably represent the most popular FAA logbook, and the two most popular EASA logbooks for professional pilots. It's really not a problem switching between EASA and FAA without changing logbook - but it's probably easiest to start as you mean to go on. That means also commencing an electronic backup - there are various commercial products to do that, or write your own; just google that topic and you'll find plenty of options and recommendations.)
Incidentally, I used one of these for 20 years (before eventually designing my own) with a mixture of CAA, JAA, EASA and FAA licences - and no great problems beyond having to change a few column headings manually... I still like it.
https://www.pooleys.com/shop/pooleys...ilots-log-book
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