Once you make your way south through the '''Kaernter Strasse''' pedestrian area again (and walk around the side streets), past the southern end and past the Opera House is the '''Naschmarkt''', a culinary market with amazing foods from all over the world.
To the east from here, about a 3/4 mile walking thru old Vienna (or taking a cab), is '''Belvedere''', a huge complex made up of palaces, gardens and parks. Since you're there in spring, you'll see the gardens in full bloom. If the place looks familiar, it's because it starred in an endless number of movies as backdrop, including James Bond.
And this is only a few pieces of downtown Vienna...