To become a lumberjack, just equip an axe and use it on a tree. Trees will only provide so much lumber, and you will need to move through forested areas moving from tree to tree. It would be wise to note that logs are extremely heavy, and that carrying a carpenters tool to turn these logs into usable boards will let you carry a bit more lumber each gathering trip.
Player vs. Player
Lumberjacking provides a bonus to damage in combat with an axe equipped. This bonus is also applied in PvM content.
Some choose to use this skill mainly for its combat bonus, but the vast majority of lumberjacks are craftsmen. Commonly found in the templates of carpenters and fletchers, lumberjacking is almost a requirement for these professions.