Yes, k needs to be incremented whether we print something or not, also p needs to go back to 2 for each new k.

k = 2;
While[k < 16, p = 2; While[Mod[k^p, 17] =!= 1, p++];
 If[p == 16, Print[k]]; k++]

A simpler way might be

In[2]:= Select[Range[16], FreeQ[Mod[#^Range[15], 17], 1] &]

Out[2]= {3, 5, 6, 7, 10, 11, 12, 14}

or just use the built-in function

In[3]:= PrimitiveRootList[17]

Out[3]= {3, 5, 6, 7, 10, 11, 12, 14}

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?