So just now i found this girl Angi south of Falkreath who is a master in Archery and gives these cool free lessons where you hit targets in a certain way and it levels your archery by a level in seconds.

bump

spoiler alert?

Or you could just recruit Faendal and have him train you, then trade with him and take the gold back.

Angi's Bow is on a weapons rack in my Dovakiin Hideout!

lol