Before proceeding with the Fallen Star quest, you should take this chance to explore the town a bit. Inside the town you can visit merchants to stock up on supplies, use artisans to craft new items, or access your stash. Interacting with NPCs after each quest is very fun, listen to their dialogs, see what they have to say.
Bron the Barkeep, in the Slaughtered Calf Inn, welcomes you to his tavern. He tells you that you can bring him whatever treasures you have no use for, and he will give you a bit so you can save up for your casket. You can buy and sell items to him, or repair your items. There is a buyback tab also, in case you sell something by accident. At this point, the only thing you can purchase is the minor health potion.
There are some villages in this town. Talking to different NPCs can get you caught up on the latest rumors. One of the villagers asks you if you had come to help them. The other adventurers who came through here didn't look so deadly. Another villager tells you that she does not care what anyone says - that thing that fell from the sky was no rock. A rock doesn't cause the dead to leave their graves.
Some villagers, when you talk to them, they have dialog choices where you can choose the topic to speak about. Talk to the villager standing near the entry. At this point you might only see one option, Tristram's Families. Select it. The villager tells you that Tristram used to be such a wonderful place to live. Her family has been here since the town grew out of the ruins of Old Tristram. But now it's horrible, the loved ones they've buried and grieved over now threaten their lives.
Exit the inn. As you keep exploring the town, note that you can also hear rumors just by walking near some NPCs. The people in this town chat and talk to each other, just like in real life. For example, you might hear a villager say that sixteen militiamen went out, and only Rumford came back, Akarat help us. Another villager responds that they should never have built on this cursed ground. Maybe Cain was right - even in death, Diablo torments us.
Look for merchants around town. For example, Tashun the Miner, standing close to where you entered town, usually will have rare items to sell you. Check with him often, in case he has new stuff in stock. He can sell interesting and powerful weapons and armor such as the Vampiric Hand Axe of Flame, Vampiric Short Sword of the Snake, Scouting Gloves, Reptilian Cloth Tunic, Gathering Cloth Pants, and Adventuring Buckler. If you are talking to him at an early stage of the game, then at this point you probably do not possess enough gold to buy any of the items.
Near Tashun the Miner, you can speak with Brother Malachi the healer to receive a blessing.
Feel free to keep exploring town and meet other NPCs, before proceeding with your current quest, The Fallen Star.