I take it that arrow coming into the top is your clock.
Have you recognized the function being performed on the serial string?
If you haven't, look closely. What would you expect if you clocked in 1, 0, 1, 0?
Since the four outputs are a function of the four bits being clocked in, what do you need to do with those input bits?
Let's say that the input/output list looked like this:
0,0,0,0 => 0 0 0 0
0,0,0,1 => 0 0 0 1
0,0,1,0 => 0 0 1 0
0,0,1,1 => 0 0 1 1
0,1,0,0 => 0 1 0 0
... => ...
What kind of gate would do that for you?