I’m not going to list out things that you can find in every travel book and/or on the Vienna tourism page. Those posting above (excluding wmt1) have done that for you already. What are your interests and likes? What one person enjoyed, you might find a snoozefest.
That said, you’re coming in December, during the Christmas markets. Focus on that! The touristy one at the Rathaus is my least favorite. Full of tat from China and too crowded. The best ones are the following:
Karlsplatz - best for crafts, by far. Lovely setting by the church. Check out the self propelled carousel and have a donut topped with sauerkraut.
Spittelberg - covering three tiny alleys, this market twinkles with charm. Make sure to go behind the stalls and into some of the buildings in this art gallery area.
Altes AKH - fairy tale setting with good food options and unique crafts.
Schonbrunn - pretty touristy, but very pretty as it is set right in front of the palace. Best market for foodies.
Belvedere - this market has the very best punsch selection. Go here after dark as the lit up Belevedere palace as a back drop is pure magic.