Why did Lyddie hesitate to enter Cutler's Tavern?

The part of the story that the question is asking about is the very beginning of Chapter 3. Luke has just dropped Lyddie off at Cutler's Tavern, and Lyddie just stands there outside of the gate. It's a great moment in the story because it is one of the few times that readers get to see Lyddie not taking action. I believe that Lyddie hesitates outside of the gate for two reasons.  

First, Lyddie is hesitant to enter the gate and Cutler's Tavern because she is intimidated. The place is huge. It's much larger than anything else that Lyddie has ever seen as this point in her life.  

Second, Lyddie knows that she is giving up her independence the moment that she walks into that tavern. During the previous winter, Lyddie and her brother had been in charge of themselves and the house; however, that house has been sold, and Lyddie has been turned into an indentured servant at the tavern. Lyddie is a fiercely independent character, and she knows that is gone once she steps through that gate. That scares her.  

Once I walk in that gate, I ain't free anymore, she thought. No matter how handsome the house, once I enter I'm a servant girl—no more than a black slave. She had been queen of the cabin...up there on the hill. But now someone else would call the tune. 

Why does Lyddie return to the tavern?

I believe that you are asking about the end of chapter 6.  

In chapter 5, Mrs. Cutler decided to take a trip to Boston in order to sell some of the maple sugar and visit her sister.  Triphena announces to Lyddie that if Mrs. Cutler can have some time away from the tavern, then Lyddie should too.  Lyddie decides to go home.  

Once she arrives at home, Lyddie discovers that a runaway slave named Ezekial is hiding out in her farm house.  The two of them talk, and Lyddie realizes that her life at the tavern is similar to slavery. Nonetheless, it is her job.  Lyddie returns to the tavern, because it is what she needs to do to pay off her family's debts.  

Unfortunately, Mrs. Cutler is already back at the tavern.  She fires Lyddie immediately.  Lyddie is not heartbroken, because now she is able to go to Lowell and become a factory girl.  

She felt more lighthearted than she had since the day Mrs. Peck brought the letter. . . "I'm going to be a factory girl, Triphena. . . I'm free.  She's set me free.  I can do anything I want. I can go to Lowell and make real money to pay off the debt so I can go home."