There are thousands of ways to lvl up.
But just remember: you are not a superhero killing thousands of enemies. You are the leader of an ARMY that kills thousands of enemies
So your priority should be getting enough money to buy some mercenary knights. They are just great to help you against unarmored bandits. If you have 4-5 knights, try to kill hroups of 15 pirates. Great loot, good exp, and not many risk to die.
When you have 15-20 mercenary knights you could be able to take down even groups of 50 pirates. Also try to find large groups of archers deserters (take a look at Nord, Vaegir and Sarranid lands). And run away from bandits/deserters from Khergit lands. You can't beat them in open field without an elite army (those mounted archers are really anoying).
That's what i do when i start a new game. Later i decide to who i want to serve and what kind of army i wanna have. But if you want to kill larger amries easy, take an army of full knights, swadian (high defence, 1h sword/hammer + shield), vaegir (full offensive. 2h weapons mostly) or sarranid (midpoint. they have 1h sword+shield+2h hammer). I prefer the sarranid mamluks, but that's for my playstile.
P.D: tournaments are good to get renown and money as Goodguy3 says, but they are hard to win if you are a beginer in this kind of combat. So save before the tournament and load the game untill you win it.