There are plenty of ways to analyze texts like The Franklin's Tale, and multiple methods are often used in one analysis. First and foremost, an analysis can't be done without reading the text fully. This will help you gain an understanding of the characters, themes, plot, etc. from start to finish. Analyzing halfway through can often cause misunderstandings or misinterpretations of something that was elaborated on later in the text.
Once you've read the book, have some questions in mind and see if you can recall parts that would answer those questions--then reread them. For example, one could ask, "What are the literary devices used throughout the book?" "What were the major plots points and how did they push the story forward?" "Were there any underlying or hidden themes throughout the book that actually had an important impact or meaning?" So on and so forth.
Use those answers to help craft your personal response to the text. A large part of analyzing a book is based on your own opinions at the end. For example, someone who loved Holden in Catcher in the Rye will develop an entirely different analysis of the text than someone who held great disdain for him. The way you feel about the characters, plot, and setting will largely affect the analysis and interpretation you have of the book.
Most analyses are also backed up with evidence from the text. As you read through, note down page numbers and sentences/quotes that could be used to back up the statements you make. It shows that there's a reason why you're saying and feeling what you are. (You can find more tips at the link below.)
For instance, let's consider the question "What important plot point happens in 'The Franklin's Tale' that is essential to the story?" From this question, we have a base for the analysis of the tale that lets us look at how one event affects the progression of the story.
For this, I would choose the husband's decision to go off and become a knight after a year of happy marriage with his wife:
A year and more lasted his blissful life
Till that the knight of which I speak of thus —
That of Kairrúd was cleped Arveragus —
Shope him to go and dwell a year or twain
In Engeland that cleped was eke Britain
To seek in armès worship and honoúr
This moment in the text signifies a life-changing event for the two of them, one that shifts the rest of the narrative as we start to follow Dorigen's sorrow. She misses her husband and wants his safe return—but is not comforted by the rocky shores. This leads her character to develop in a specific way, often sad, and eventually giving into her friends as they try to help. She begins to question God out of fear for her husband returning to the rocky coast:
But wouldè God that all these rockès black
Were sunken into hellè for his sake!
These rockès slay mine heartè for the fear!
The development of Dorigen's character (and situation) all hinges on the fact that her husband leaves to pursue knighthood. Because her friends try taking her to a dance in a garden to cheer her up, another man starts to fall in love with her:
Unwitting of this Dorigen at all,
This lusty squire, servant to Venus—
Which that y-cleped was Aurelius—
Had loved her best of any creäture
So, you can start to see how "The Franklin's Tale" is largely affected and progressed by the initial decision for the husband to leave for two years to become a knight. As you go through the rest of the tale, you can find other textual evidence that leads back to this one essential plot point, as well as other examples of how Dorigen's character is affected and developed because of the plot point.
And although the husband returns safely, Dorigen has already mistakenly agreed to be Aurelius's lover if he removes every rock from the shore (a promise she made thinking it would help her husband return safely, not realizing the consequences that would come from it). That decision in and of itself leads to the development of Aurelius and his role in the tale, so we can see how one major plot point has the potential to affect the progression of the rest of the story.