Instructions
Complete all quests you come across. Hundreds of hours worth of quests are available for you to complete, all of which provide experience and help you advance in level. Avoid completing the main quest before the side quests if you haven't yet bought the expansion pack; the base game does not let you continue the game after the final mission.
Defeat as many enemies as you can. For best results, pick fights with enemies that are a little bit stronger than you and are a little higher in level. For example, deathclaws are generally a tough fight until you are level 20. A general rule to follow is if you can win a battle and nearly die, you can get a very sizable amount of experience from the combat.
Purchase the "Broken Steel" expansion pack. This is available on the PlayStation Store for $9.99 as of July 2011.