Once Emily tells Nancy about Josiah Crowley, you can end the conversation and leave. That's not a good idea! You have to talk to her about Jim Archer, or else you can't meet him at the Main Street Bank.
This conversation is simple. Emily is reluctant to visit Mr. Archer. Nancy can say, "Not one of your favorite people, huh?" or "You don't sound very happy about it." If you pick the first one, Emily assures you that he's nice, before complaining she's bad at business. If you pick the second one, Emily complains she's bad at business. It's about 4.5 seconds faster to pick the second option.
Emily will ask Nancy to visit in her place. Nancy can say "Sure" or "I guess". She says "I guess" slowly, so the faster option is "sure". It's about .2 seconds faster. Then you can end the conversation, and you're finished with all the conversations at the start of the game!
Of course, what you do after the conversations is go directly to Topham's, so you can meet him and have a conversation. I guess the start of game conversations are not over.
I think there are two other things you can do at this point.
1. You can do the clock puzzle in the den at this point. You could do it earlier, after talking to Jane. Or you can do it later, when you go back to the inn to ask Emily what her mother's middle name was. There's not really a time that's best for solving the clock puzzle in the den. You can do it whenever. Similarly, there's not really a time that's best for asking Emily what her mother's middle name is. You can do it before or after your first trip into town; you just need to have it done, by the time you're ready to go into the carriage house.
2. Minigolf! You can do minigolf before talking to Topham or after talking to Topham. Since minigolf is a difficult/random challenge, it's probably best to put it here, as close to the start of the run as possible. That way, if you really screw up at minigolf, restarting the run from the beginning again is less of a disaster.