As there are well over a dozen markets in Vienna, I could understand the confusion of referring to "the market." These days, Naschmarkt is for the tourists and many stalls are now selling trinkets, which is very disappointing to say the least. For a more authentic market, I would suggest Karmelitermarkt on a Saturday morning or Brunnenmarkt. I also have a soft spot for Rochusmarkt, my local market, on Fridays and Saturdays. It is small but might with local farmers and some interesting places for a bite to eat.
I have previously provided tips on cafes, but to summarize, go to Diglas for cakes, go to Demel for the elegance, go to Gerstner for the strudel, go to Sperl for what you think a cafe should look like, go to Vollpension for the best overall grandma cake experience.
For restaurants, I would recommend Plachutta for Tafelspitz, a wonderful iconic Viennese dish, Huth for schnitzel, Bitzinger with the pink rabbit for Wurstel, Palmenhaus for a sunny lunch, Rebhuhn for an excellent local meal (walk down the nearby Servitengasse for some classic ambiance.
For heurigen, Schuebel-Auer is my favorite for food for certain, followed by Feuerwehr Wagner. I also really like the Stammersdorf area, Wieninger in particular as their food is outstanding.
Jazz - I have no idea.
Churches - Karlskirche, Kirche am Steinhof, Stephansdom
Museums - Really depends on your taste. Any good guidebook or internet search will give a good overview. My personal favorites are the KHM and the Belvedere.
Parks - Oberlaa, Volksgarten (for the roses), Augarten, Stadtpark
Must-sees/gems - No idea as I don't know your interests. Time of year would also be helpful.