They played 25 matches in two months, travelling from coast to coast, winning the first 24 but ending up with a draw against Fall River. Full details can be found here. The US Football Association presented the tourists with an attractive trophy (pictured) which has the SFA crest as well as the Third Lanark badge, either side of the US logo.
The players sailed out on the Cunard liner Cameronia, leaving Liverpool on 11 May 1921, and the passenger list (below) is full of interesting detail about the players, some of whom gave their profession as footballer, while others gave their 'day job' such as publican or joiner.
With a bit more digging, I've also found similar lists for the first 'official' SFA tours across the Atlantic, in 1927 and 1935, as well as various club lists including Third Lanark's venture to South American in 1923. These are great examples of the kind of material that is now available to historians after many years being hidden from view.