Where am I going wrong with my code?

You've got lots of basic syntax error. May be you should start with small example first in order to learn the basic syntax., You need ";" to separate statements in M. You can't write

k = 2;
p = 2
While[k < 17,
 p = 1
   While[GF[17][{k}]^p =!= 1,
    p ++
    ]
   If[p == 16, 
    Print [k]
     Break[]
    ];
 k++
 ]

notice there is no ";" after the p=1 and no ";" after the ] there., You also need ";" after the Print.

It is important to know the difference between "," and ";" in Mathematica. I have not tried to run your code, just an observation. Also the GF[17][{k}]^p looks strange but I never used this function. Have you tried to test this on its own first to make sure you get the syntax correct?