"...special...landmarks that you feel cannot be missed." How deep is your interest in the history, beyond superficial view? Some to track down, ie these historical sites. As it is landmarks you're after, I'll leave out the museums.
I suggest...a. the Heldenplatz, where Hitler addressed the crowds after the Anschluss (annexation), where over a quarter of a million packed in to see him.
b.. Maria-Theresien Platz..the Empress surrounded by her diplomats (main one is Kaunitz), and military commanders, eg, the defensive commander Daun
c. Schwarzenbergplatz.. to see the Soviet war memorial at one end and, at the opposite end, the statue of the Habsburg military commander Schwarzenberg ( he still lost to Napoleon a few times). Other Habsburg military commander statues to be seen in Vienna are the big one of Radetsky, in front of the big post office building, Prinz Eugen in front of the National Bibliothek (Library) and opposite him across the way is that of the Archduke Charles (Karl), that of the Archduke Albrecht at the Albertina.
d. . the Deutschmeisterplatz ...the biggest Habsburg military soldiers' statues in Vienna along with the Landwehr (militia) memorial just down from Westbahnhof.
e. . the Zenrtralfriedhof where the statues of the composers and the WW1 and 2 war cemeteries can be seen.
f. The Holocaust memorial across from the Albertina.
g. . There are 2 memorials of K. Lueger, mayor of Vienna at the turn of the century, and Hitler's favourite mayor, one is across from Westbahnhof near Mariahilferstrasse, the other is a statue in the 1st district.
h.. The statues of Goethe along the Ringstrasse, Germany's greatest poet, and that of Grillparzer, the literary master in Austrian lit at Belvedere.