The is a determiner, first is a number, professional is an adjective, baseball is an adjective, league is a noun, was is a verb, later is an adverb, started is a verb, in is a preposition, 1871 is a number.
"The" is an article. "First professional" is a compound adjective. "Baseball" is an adjective too because it's describing the league. "League" is a noun. Was is a verb. "Later" is an adverb. "Started" is a verb. "In" is preposition. "1871" is an adverb.
