I know Vilkas trains two-handed weapons. Who's the equivalent of Vilkas for one-handed weapons?
In Whiterun, you can meet a guy called Amren, who will happily train you in one-handed combat at a Journeyman level for a fee.
Athis, a member of the Companions, also in Whiterun, will train you at an Expert level.
Burguk of Dushnikh Yal will train you at a Master level. You can find him in The Reach.
Source (for this and other trainers): Skyrim guide
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The best way to get one-handed for free to is:
Then you can take anyone in the companion "guild" as a companion. This means that you can train 5 levels of a skill per "major level" and take your money back by talking to your companion and saying: "I need to trade something."
You will therefore find your money.