Your build for the Barbarian class is highly determined by the quests you can undertake. Hacking and slashing your way in the game just doesn't work anymore as it did for Diablo 2.
For starters, do not grind the same monsters or do boss runs over and over again. It produces diminishing returns and Blizzard has set the game up in such a way that the rewards look better if something is done differently. Venturing out is a very good option.
A Barbarian is a class that excels when they are in a team. Because most of their attacks are melee, they function well as a tank but are better off with characters to back them up with range attacks and provide support. If you are a low level barbarian, make sure you pair up with someone that provides a lot of healing spells and this will keep your game going longer and stronger.
One of the biggest temptations of a barbarian is to grind through the same pack of mobs over and over again. This might work well in the previous game, but the way the game is setup requires you to diversify your strategy rather than attack the same boss over and over again.
A barbarian must not sit still. Its primary skill is called fury which increases as they deal damage and take damage. Do not wait for your allies to deal the damage as this will hamper your ability to level up. The fury ability also unleashes a huge range of skills that will propel the Barbarian's damage as well.
Last but not least it is very important to experiment with rune stones. Rune stones diversify the play of the Barbarian greatly and don't be afraid to try out new things as the damage dealt and mobility can be greatly affected.