The
Library of Congress (LOC) is the de facto place in the U.S. to check original publishing info, but it doesn't include British publications (LOTR was first published in England, in 1954-55). In LOTR's case, there have been a myriad of versions since the original, so it would take some detective work to figure out what was the true first edition for a book like this. But normally, the LOC is your best bet to start with.
The majority of books published, however, are only published once...so the dates at Amazon/BN, etc., are good indicates most of the time. It used to be that you could trust a hardcover to be the first issue of a book, followed later by a paperback...but that's not the case any more.
Cheers,