A New Beginning is a quest. It is triggered after you fought the bandit that attacks you as soon as you start a new game and consists in talking to some residents of The Weeping Town.
Starting the quest
Fighting the bandit
As you are thrown into the game you will be attacked by a single bandit, his weaponry varies and he can be a archer or a melee fighter, in any case, a couple of hits should be enough to kill him. Although it's not necessary to win, it will give you some experience and weapon proficiency.
After the fight, no matter the result, you'll find yourself inside Osbert Clerk's house, go talk to him and he'll tell you his plans of leaving Westeros for Essos. He'll advise you to do the same. At this point you'll have to take one of two answers:
I think i'll stay [...]. Starts the quest A New Beginning and he'll give you 200 coins.
Go to Essos, what do I care? Finishes the conversation and don't start the quest A New Beginning.
Choose the first to begin the quest.
Speaking to Septon Garibald
Enter The Weeping Town and go to the castle, Septon Garibald is dressed all in white, talk to him and you'll ask about work. He'll say that he has nothing right now, nonetheless he'll point you to three tasks. The first is to help some villagers deal with a wight, the villager is named Goodman Tom and he's at the tavern, you can go to him to complete the quest Terror in the Cairn. The second is to deliver a letter for Ser Clifford Gower, this starts the quest The tourney at Ninestars. The third and final task is to deliver a letter from Garibald to the Archmaester Haradon at the Citadel in Oldtown.
Accept this to proceed to part two of this mission and get 200 exp, also it'll automatically start the quest The Knights of the Mind.
Speaking to Ser Clifford Gower
Leave the castle and then come back immediately and Ser Clifford Gower will appear, talk to him. He'll explain that he's been invited to a tourney but cannot attend and asks you to deliver a letter and a gift in his stead.
Accept this to start The tourney at Ninestars, you'll get a reward of 150 coins and the quest will be finished.