In the sentence "They are always late to the ball game.", what part of speech is "always" and "late"?

Answer 1

always should be an adverb...describing what the action is which is the verb "late". Try that, it "should" be correct.

Answer 2

Yes, "always" and "late" are adverbs.
http://www.uottawa.ca/academic/arts/writcent/hypergrammar/partsp.html