I'll ignore using nest, you can add that later.

First you want to remove the semicolon and print so that {i,xn,f(xn),abs(xn-xn-1)} is the last statement in the loop before last right bracket. By doing so the Block will exit with the statement as Out[]. Also, you can use Table instead of While: avoid While.

Now you have a block exit with a list of lists. {{list},{list},...} which can be accessed with Part[list,spec].

You can re-arrange the list with Partition[list,spec] if you need it in a different grouping.

I think I know what you are talking about, but I thought you need while though. While you have a set number of iterations do so and so. I read about Newton's Method. I will try what you said, but can AppendTo work? My professor said to create an empty list and append what I have in the loop to the list. I tried doing that, but I got nothing.