Too bad you aren't in town this weekend. It was in the 50s and sunny, which is very, very unusual for this time of year. Looking at the extended forecast, however, it looks like next weekend might be nice too.
So here is what I would do:
Friday: Walk around the first district; start at the Opera and walk up Kaerntnerstrasse to Stephansplatz. Check out the cathedral; walk down the Graben, turn left o Kohlmarkt. Stop in Cafe Central on Herrengasse; walk through the Hofburg; cross the street and walk the Ring in the direction of the Rathaus. Once at the Rathaus, rent a pair of ice skates and make a few laps on the ice path, which runs through the trees and garden. Atmosphere is great. Get on Tram D in the direction of Nussdorf; take it to the end and have dinner at Scheubel-Auer, a heuriger (never heard oomph-pa music at one of these as that is really an alpine/bavarian thing).
Saturday: Go to the Naschmarkt for breakfast. I like Neni. Explore the market and be sure to check out the building in the area, such as the Majolikahaus and the Secession building. Art Nouveau Marvels. Go have a coffee at Cafe Sperl. Take the U4 to Schonbrunn and explore the gardens and palace. Not really necessary to go inside, in my opinion. Have dinner at Figlmuller, Huth or Palmenhaus; go to the Opera.
Sunday: Go to the Belvedere Palace/Museum; see the Klimt pieces at least. On the way back into town look at the Karlskirche.