I'm having a horrible time trying to get to the other cities. I seem to be unable to break the quest needed to continue; for example, the man and boy who damaged the scarf will not appear after I fixed the scarf.
Also, I am not able to advance to the 2nd Chapter cities.
Any advice would be appreciated :)

According to the Walkthrough:
https://www.ign.com/articles/2003/03/19/prince-of-qin-walkthroughfaq-390164

" Old man and Ming'er (outside in the southwestern part of city): Talk to the
old man and he'll tell you how Ming'er damaged a scarf left to him by his
mother. Agree to help them, and you'll receive 800 experience points. You
can fix the scarf at the embroideress' house in the northern part of the
city. Tell the embroideress the reason you need the scarf repaired so
she'll only charge you 100 gold (otherwise it'll cost 500 gold) and then
walk all the way across the city to talk to the old man again. Then walk
across the city to fetch the repaired scarf, and then walk all the way back
to give the scarf to the old man. Your reward for all that walking? 1000
experience points and 1 fame point. "

If this doesn't work then you've probably encountered a game bug and should try to restore a previous savegame (from before starting the quest) if available.