To go after someone or something; to pursue or follow, or to continue in the same direction or manner.
Compound of 'after' (Old English 'æfter') and 'go' (Old English 'gan', from Proto-Germanic 'ganan'). Archaic English formation for 'pursue' or 'follow'.
This word fell out of use when English speakers moved toward using 'follow' and 'pursue,' but it shows how productive English was at building new verbs—you could just add 'after' to almost any motion verb to mean pursuing something.
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