What significant decision does Lyddie's mother make in chapter 1?

Lyddie's mother actually makes two momentous decisions in chapter 1. First of all, she decides to move the family to her sister's house in Poultney. This is because Mama has got it into her head that the end of the world is coming. She's been spooked by the unwelcome appearance of the bear in the family home and has taken it as some kind of dark prophecy. As Mama's sister, Lyddie's Aunt Clarissa, is a deeply religious woman, Mama feels that she and her children would be better off staying with her in Poultney.

However, there's not enough money available for the coach fare, so Lyddie and her brother, Charles, stay behind while Mama and the other siblings head off to Aunt Clarissa's. Throughout the following winter, Lyddie and Charles make a good job of running the farm without adult supervision. Despite having no money and almost no supplies, they manage to survive a particularly harsh winter.

But in the springtime, they receive some very bad news. Mama has made another momentous decision, and it's not good for either Lyddie or Charles. She's hired out Lyddie to work at Cutler's Tavern and Charles at Baker's Mill in order to raise some much-needed cash. For good measure, she's also lent the family's pasture fields and sugar bush to Mr. Westcott to repay debts.

In chapter 14, what decision does Lyddie make?

Lyddie makes several decisions in chapter 14, so I am not one hundred percent sure which decision that you are asking about.  

One decision that Lyddie makes is the decision to not sign the petition for improved work conditions.  She is afraid of getting blacklisted from the factories. 

In chapter 14, Lyddie is now the factory's best worker.  She is working multiple looms at the same time. Because of her proficiency, Lyddie is assigned to train Brigid, a new factory girl.  Lyddie thinks that Brigid is stupid and incapable, and by the end of the first day, Lyddie decides to tell her boss that Brigid is ready for her own loom.  Lyddie does that because she is frustrated with teaching the new girl.  

By the end of the first day, the girl was far from ready to operate her own machine, but Lyddie had run out of patience.  She told Mr. Marsden to assign the girl a loom next to her own. 

The last thing that Lyddie decides to do is send a dollar home to her mother to help pay off the family debt.  

What decision does Betsy make in "Lyddie" and why?

The question is a bit broad, because a specific chapter is not named. Betsy makes decisions throughout her time in the novel, but I believe that you are asking about her decisions in chapter thirteen.  

In chapter thirteen, Betsy decides to sign the petition for better working conditions in the factories. The current conditions are atrocious. Long hours, minimal breaks, dangerous machines, volumes so loud that going deaf is a real possibility, and fibers so thick in the air that lung problems are almost guaranteed. Signing the petition sounds like the smart thing to do, but Betsy is risking her job by signing the petition. She knows that she can be fired for signing it, and she knows that she can be blacklisted from all of the other factories as well. Nevertheless, Betsy signs the petition . . . and then gets fired.  

Betsy's final decision in the chapter is to apply to attend Oberlin College.