Arriving at Vienna Hauptbahnhof - best way to get to City?

Vienna Hauptbahnhof is further from the centre than most European big-city stations, but not much further.

The obvious answer is the U-Bahn (metro, Underground) into the city cnetre.
A U-Bahn map is here https://www.wienerlinien.at/media/files/2019/svp_300910.pdf
The red U1 line is 3 stops from the Hauptbahnhof to Stephansplatz, which is the absolute city centre
A map of the city centre is here: https://www.wienerlinien.at/media/files/2017/city-plan_220937.pdf

Schönbrunn is in the suburbs. It is a stop on the green U4 line, you can see it on the map.

The S7 to the airport does not run from the Hauptbahnhof. Ideally you want to go Hauptbahnhof - (U1) Stephansplatz, then U1+U4 to Schönbrunn, then U4 to Mitte and S7 to "Flughafen Wien" (bottom right on map).
Leaving your bags at the Hauptbahnhof would involve a detour back to there. Not sure of the best solution.

Instead of the S7, save time and spend a bit more money and take the RailJet from Hauptbahnhof to the airport. Yes, it is more expensive than the S7, but is nonstop in 15 minutes and goes twice per hour at 12 and 40 past the hour. Cost is about 18 EUR for 2.

I don't know if there is luggage storage at Wien Mitte. If there is, you could take the S-Bahn from Hauptbahnhof to Mitte, store bags there then take U-Bahn to Schonbrunn, tour the city center, return to Mitte to get the S7 to the airport.

Instead of the S7, save time and spend a bit more money and take the RailJet from Hauptbahnhof to the airport. Yes, it is more expensive than the S7, but is nonstop in 15 minutes and goes twice per hour at 12 and 40 past the hour. Cost is about 18 EUR for 2.

Sorry, but this is partially incorrect.

The fare for both the RailJet and the S7 is the same, namely 4.20 Euros. The CAT (City Airport Train) is the expensive rip-off which you should avoid (single €11, return €19). The CAT runs the same route as the S7, but is 5 minutes faster.

The ticket for RailJet or S7 permits continuing your journey on the metro, whereas the CAT ticket does not.

There is a train from the airport to the city every 15 minutes, RailJet and S7 alternating. It is a waste of time to wait for a particular train, just take next one. If it is a RailJet go to Wien Hbf and take the U1 (red line) to the very center (St. Stephen's), if it is a S7 go to Wien Mitte and take the U3 (orange line). U1 and U3 intersect at St. Stephen's Square.

Although the S7 is slower than the RailJet, you are still faster in the center than when waiting for the next RailJet.

Both U1 and U3 intersect with the U4 (green line) which will bring you to Schönbrunn palace.

Thanks wmt1. Good to know. I was going off the pricing from www.oebb.at.

The tram route D starts from the Hauptbahnhof, then heads for the Ring and travels anticlockwise around it from the Opera past the Hofburg and the big museums. It's not much slower and a lot more scenic than the U-Bahn. If you want to head straight to Schonbrunn the U-Bahn is a better choice.

Tram D stops at Hauptbahnhof Ost (East) at the very end of all platforms, not easily accessible if you are already in the main hall. Coming from Hauptbahnhof line D runs clockwise around the Ring, but only between Schwarzenbergplatz and Börse (former stock exchange).

If you want to make a full trip around the Ring boulevard, take U1 from Hauptbahnhof to Schwedenplatz and start from there.

You have two choices: the RingTram, a yellow tram for tourists with audio explanations, making a full turn around the Ring; or you take ordinary trams for a full round. In the latter case take line 2 clockwise to the parliament, change there to line 1 and complete the circle to Schwedenplatz.

Just curious, what would a taxi cost? Maybe 50 Euro each way? / 2 riders = 25 euro per person each way. Just guessing. Faster???

Just curious, what would a taxi cost?

A taxi from where to where?

A taxi from the airport to the city is €30 when prebooked. And it is not faster than the train.