That would be great Hanto. Also, cheers Ginner!
So places we ate/saw that I can recommend are:
Massive in Viennese culture. The idea is you buy your coffee, and then you essentially can stay as long as you like, e.g. 3 hours!
Demel - Very famous and does amazing cakes and coffee. This is also near the Hofburg, which is the Bond Street for shopping. Worth taking a horse carriage ride if you want to get a look around.
Gerstner K. u. K. Hofzuckerbäcker - Similar, very good cakes as well. Can recommend the strudel. If you want something like custard, it is called vanilla sauce and don't ask for anything otherwise!
Kleines Cafe - More bohemian than the other too. As you can smoke indoors, we didn't stay very long.
Cafe Central - I didn't get a chance to visit this, but it is near Placutta, so could be worth a visit.
Plenty of hot dog stands, so can't go wrong
Buffet Trzesniewski Dorotheergasse - Great sandiwches and small beers, the name of which i've forgotten, but sounds like 'fifth' and is traditional pilsner, so not too heavy.
Plachutta - Traditional place for things like schnitzel. I had the tafelspitz, which is great but quite rich! Apparently its Putin's favourite restaurant.https://www.plachutta-wollzeile.at/
Trattoria Toscana La No - I think this is the place we went to, but it looks like it moved. If you do go, ask them if they use to be on a street called Dorotheergasse. If yes, then its the same place. Great Italian food.https://www.tripadvisor.com/Restaurant_Review-g190454-d8499358-Reviews-Trattoria_Toscana_La_No-Vienna.html
Didn;t go in that many shops but for sneakers, you have Solebox
If you want a bit more traditional/regional:
Muhlbauer - The latter two are stocked by Trunk, so gives you and idea of the clientele
Spanish Riding School is very impressive, as is the State Opera house
The Kunstmuseum is a great gallery. Mainly went for the Vermeer they had there, but there are a fair few famous paintings in it.
If you want Klimt etc. they are in another gallery on the other side of the city.
Schönbrunn Palace is very good as well. Great views etc. Need to use the metro to get there though.
The zoo is there as well as is worth a visit. They have Giant Pandas if you want to see that!
Prater amusement park - The famous Ferris Wheel is here, but it essentially has a pop up equivalent of Tivoli next to it. It is like a travelers fun fair on steroids! Great fun to be fair.
Saint Stephens Cathedral. The tour of the catacombs is quite impressive
If I think of anymore, I'll add them. The best thing to do, which is true of any city, is walk as much as possible, you'll get a much better experience. Hope that helps!