"First world" was a Cold War era term, where "first world" is America-aligned nations (so by definition, that would include the U.S.), "second world" was Soviet-sligned (no longer really in use since the Soviets no longer exist), and third world was everybody who didn't pick a side (so China after Sino-Soviet split, India, Brazil, etc.)

Most organisations use the more neutral terms "developed" and "developing", and the U.S. is definitely a developed nation (there are less binary terms that economists and such use to categorise even further that you could look into, just check the organisation who runs it since many can be biased towards certain things compared to others)