2Pi is multiplied by 1.5 z - z^2, and this is indicated by either explicitly by 2 Pi*(1.5 z - z^2) or implicitly by juxtaposition 2 Pi(1.5 z - z^2). The result is the argument of Cos, and goes within square brackets: Cos[2 Pi (1.5 z - z^2)]. Altogether

f[z_] := 
 Exp[-z] Sin[2 Pi (0.61 z - 0.22 z^2 + 2 Cos[2 Pi (1.5 z - z^2)])]

Check again your syntax. The argument of functions like Sin and Cos goes inside square brackets. Pi is not a function, it is a numerical constant, it is not followed by [].

The underscore z_ indicates a pattern, and belongs to the left-hand side. Square brackets are for the argument of functions, curly braces for lists, round parentheses to indicate the order of operations. It is VERY coherent and worth internalizing the distinction, which is lost to traditional math notation:

g[z_] := Cos[2 Pi (1.5 z - z^2)];
f[z_] := Exp[-z] Sin[2 Pi (0.61 z - 0.22 z^2 + 2 g[z])];
Plot[f[z], {z, 0, 2}]