You will naturally earn level 20 by killing enemies and doing side quest.
I would highly suggest doing any and all side quest you get as this will increase your level quicker. There is also a series of challenges you can do early in the game where you kill so many enemies in less than 7 seconds from each other. So you kill one enemy then kill another one in under 7 seconds then repeat that. These are those challenges.
Relentless Kill 5 enemies in succession with no more than 7 seconds in between each kill 500 XP
Chain Killer Kill 10 enemies in succession with no more than 7 seconds in between each kill 1000 XP
Killing Spree Kill 15 enemies in succession with no more than 7 seconds in between each kill 2000 XP
Conveyor of Death Kill 25 enemies in succession with no more than 7 seconds in between each kill
5000 XP
In total there is 8500 XP in just one set of challenges which can take you up a level or two.
The best place to do this challenge is in skag gully. Just run around and kill everything as fast as you can and shoot the raks to spawn more of them to easily complete the challenge. A shotgun or high mag assault rifle will do the trick.
Happy Adventuring Vault Hunter!