Where can I train 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

The best way to get one-handed for free to is:

• First finish the companions quest.

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.