The Rangers, to their second consecutive World Series. Congratulations
The last team to lose and then win a WS in consecutive years? That would be the 1988 and 1989 A's. Before that, you got back to the 1976/1977 Yankees. The last team to lose twice in a row? The 1991 and 1992 Braves. An American League team? Well, the 1963 and 1964 Yankees, but they had just won the previous two years.
In the history of the American League, until now only the Yankees, Blue Jays, A's, Orioles, Tigers, Twins (as the Senators), and Red Sox had previously won back-to-back pennants. Oddly enough, all but the A's and Orioles did it the first two times they ever won the pennant. So the Rangers are the eight AL team to join the club.
In looking that up, I learned something I don't think I ever knew. You all know that Philadelphia won the 1905 pennant and then were clobbered by Christy Mathewson and the Giants in the famously deadball WS, right? What I did not know is that the Athletics won the same number of games as the White Sox that year, but played four fewer games; had the rules required everyone to play out the schedule, Chicago would have had two games remaining and Philadelphia 6.
Getting back to consecutive pennants, all 8 of the NL franchises that began the 20th century have turned that trick, although the Phillies hadn't done it before the current team in 2009-2010. However, none of the NL expansion teams have done it, while this is the second AL expansion team to make it.
There has never been a WS between two expansion teams.