The Chicago White Sox are their biggest division rivals, but the Twins' main postseason rival is definitely the New York Yankees - they ousted Minnesota from the Division Series three times in the 2000's - in 2003, 2004, and 2009.
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Probably the New York Mets. Or the Cubs, they seem to not have gotten along lately.
The main rivals of the Marlins are the New York Mets and the Philidelphia Phillies.
main rival now a days would be tigers. but the Indians real rival is the New York Yankees.
I would say that would be the Angels. At least when the Mariners used to be competitive.