Even if you do not like to follow a guide, it helps when you do not where to go next.
31/20/0 Arms is best for leveling. It provides damage value on every tier compared to Fury, which only yields cruelty on the 1st tier and has a drought till the 4th tier (enrage). Arms also includes tactical-mastery which is great for tanking in dungeons and world PvP.
First 31 points into Arms:
Then the last 20 points into Fury:
Axe spec is always preferred while leveling, even if you do not always have an axe. You have some freedom with the Tactical Mastery, Two-Handed Weapon Specialization, Improved Charge and Improved Hamstring talents (indicated by *). If you do not (need to) kite or you play on a PvE server, you may want to skip improved-hamstring completely. In the Fury tree, you may also want to substitute Booming Voice with unbridled-wrath and Blood Craze with more points in Improved Battle Shout (again indicated by *).
Do not train all class skills from the trainer while leveling! The primary objective is to have enough gold at lvl40 to buy the normal 60% mount.
You may also want to improve your gear, make some gold and optimize your supply chain.