Collect[ExpandAll[
  Simplify[((((((z^2 - qz - 1/27 p^3)/z) /. z -> w^3) /. 
          w -> 1/6 (3 y + Sqrt[3] Sqrt[4 p + 3 y^2])) /. 
        p -> (b - a^2/3)) /. q -> ((-2 a^3 + 9 ab - 27 c)/27)) /. 
    y -> (x + a/3)]], x]

You are right, Collect[] seems to be enough

In[2]:= Product[x - o, {o, 0, 9}]
Out[2]= (-9 + x) (-8 + x) (-7 + x) (-6 + x) (-5 + x) (-4 + x) (-3 + x) (-2 + x) (-1 + x) x

In[3]:= Collect[Product[x - o, {o, 0, 9}], x]
Out[3]= -362880 x + 1026576 x^2 - 1172700 x^3 + 723680 x^4 - 
         269325 x^5 + 63273 x^6 - 9450 x^7 + 870 x^8 - 45 x^9 + x^10

So how would I simplify and collect

What do you mean by how? Use Mathematica as mentioned above. If you don't have Mathematica at your disposal it will be a rather long and tedious piece of substantial algebra to do - that's why Wolfram inventend Mathematica.