Question. When do I use the equal sign = by itself and when do I use := including the colon?

Answer. The general rule (to which of course there are exceptions) is:

• Use Set (=) to assign constant values (including constant vector values) to names.
• Use SetDelayed (:=) to define functions.

Examples:

myscalar = 22/3;
myvect = {2, -5, 3};
f[x_] := x²
g[x_, y_] := 2 x - 5 y
reflect[v_] := {v[[1]], -v[[2]]}
translate[pt_, t_] := pt + t {2, 3}
newtriangle[fig_, t_] :=
   image[translate[#, t]&,   fig]

Notice that SetDelayed (:=) typically goes with definitions in which pattern variables (names followed by underscores) are used on the left side of the definitions.

Be sure to type := with no space between the colon and equal sign! (The two symbols together constitute an abbreviation for the named built-in function SetDelayed.)