After you are done trading and exploring the town a bit, it is time to continue with the main quest:
The Fallen Star
Objectives: Talk to Captain Rumford at the gate of New Tristram.
Head out of the gates and speak to Captain Rumford. You ask him how you can help fight the Risen Dead. Rumford tells you he admires your courage, but that Captain Daltyn and the militia were slaughtered by those things, only Rumford survived. He has seen you fight though. He wants you to strike at the wretched mothers and their queen. They are the ones vomiting out those horrors.
You receive a new objective:
Kill the Wretched Mother
Suddenly a loud bang is heard. The Risen are attacking the barricades. You receive your next objective in this quest: to Kill the Wretched Mother. Attack the Risen near the barricade. You kill the Wretched Mother and complete that objective, but Rumford tells you there are more of them in the ruins down the road.
Your objectives now are:
Kill the Wretched Queen in the Old Ruins
BONUS: Wretched Mothers Killed 0/3
The bonus objective is not a mandatory objective, you can complete the quest without doing it. However, you can get some extra experience from doing bonus missions, and most of them take place while doing the main quests anyway, so why not?
You can usually speak to the quest givers again to have them repeat the quest description. For example, talk to Rumford and he will tell you again about the Wretched Mothers and their Queen. You can also ask him about Captain Daltyn. He tells you that when Leah told them the dead were pouring out of the cathedral, they quickly went to put an end to it. At first it seemed they were succeeding, but the Risen just kept coming until the militia were overcome. Captain Daltyn and the men fought valiantly. They protected Rumford. He was a farmer, not a soldier, and should not have been out there with them. He does not know how he made it back to town, none of the others did. And now, somehow he is supposed to lead the Militia.
As the main story progresses, and as you talk to NPCs, you unlock additional dialog choices, so it is always recommended to check with the NPCs occasionally to see what new dialog options they have available.