Vienna Public Transport Flex Pass

First the basics:

Download the WienMobil app onto your smartphone to get all information about public transport, including the purchase of tickets. The cheapest and most versatile ticket is the Flexible 7-Days-Pass for €17.10 which allows you day and night unlimited use of all means of transport (trams, subways, busses, S-trains) for a week starting on the day of your choice.
https://www.wienerlinien.at/web/wl-en/wienmobil-features

All public transport passes are valid on the territory of the State of Vienna. The airport is just a few miles outside of Vienna, therefore you need an add-on ticket (€2.10) to your pass when going there.

The CAT, which is a rip-off, is NOT included in Vienna's transport passes.

There are no turnstiles, you just hop on and hop off, and you must have your transport pass with you, either on paper or on your smartphone.

For finding the best way from A to B (including alternatives) I prefer the VOR AnachB App because it gives you the routes not only using public transport, but by car, bicycle and walking as well.
https://www.vor.at/en/timetables-mobility/vor-apps/vor-anachb-app

Edit:
Effective July 1st, 2024 Wiener Linien, the operator of public transport in Vienna, will change some of its ticket offerings.
Three items are of interest for tourists:

(1) The calendar week ticket (Mon-Sun) will be abandoned.
(2) The flexible 7-Days-Pass, as described above, i.e. purchased online, strictly personal, will cost €19.70 now.
(3) The new flexible 7-Days-Pass, if purchased as a paper ticket, is transferable, will cost €22.60 and can be obtained from ticket vending machines, ticket counters, tobacco shops.

wmt1 gave you good advice above. Essentially, the public transportation system in Vienna is an honor system, meaning that you do not show tickets to enter any means of public transportation as it is assumed that you have paid already (that you are honest).

That said, your pass will include the validity dates of use, so make sure you are using it on those dates.

In addition, the public transport authority will do spot check on buses, trams, regional trains and on the U-bahn, to ensure compliance. I have been spot checked probably 5 times in the last year - and I ride multiple forms of public transport every day. Public transport workers conducting the checks will be in plain clothes, but they will have official badges identifying them, so please do not think this is a scam - it is not. I've actually seen the plain clothes checkers get very upset with tourists who wave them off/ignore them thinking they are asking for money or scamming (activities which are already uncommon in Vienna anyway). The checks may or may not speak English.

.. S7 train from the airport?

There are different trains leaving from the airport going to the city. Which one is the best depends on where you want to go to in Vienna.

The RJ/RJX trains (RailJet) go to Vienna's main train station (Hauptbahnhof), where you have connections to subway U1, S-trains, etc. The ride is only 15 minutes.

The S7 (or REX7) trains go to Wien Mitte station (connection U3 and U4), Praterstern (U1, U2) and Floridsdorf (U6).

We will take the train to Wien Mitte.

What is the name of your hotel?

hello, I'll be in Vienna for 2 weeks starting next Saturday. How do I add on the extra ticket to and from the airport to my 7 day pass? I will probably end up getting 2 7 day passes.

How do I add on the extra ticket to and from the airport to my 7 day pass?

It is an extra railway ticket, best to be bought at a ticket vending machine at the railway station. There is a button to tap on the screen for entering a fare reduction. From this (long) list you must choose that you have already a ticket for the Vienna city area.
The ticket is €2.10 then instead of €4.50 regular fare. (This is not true for the private CAT train.)

DD and I will head to Vienna for the Christmas markets mid-Dec (yay!). Planning to buy the 7 day Wienerlinen pass via the app. I understand the OBB ticket for the S7 train is separate, and is a little over 2EU each (2.10EU?), but I cannot figure out if there is a way to buy it online at OBB (just the fare from VIE to city limits). However, when I do a booking for 2 adults to Praterstern (our station) it quotes me 4.60EU, so I'm thinking the difference is worth it for the convenience of having the tickets in hand when I get off the plane. I do see that there are REX trains and S7 trains running at that time. Can I use a REX ticket on the S7 train or vice versa? Thanks for all guidance.

I was looking for this information too, thank you! I prefer paper tickets- are the kiosks easy to find/use at Hauptbahnhof? Just buy, print, and keep with you, nothing else?

I prefer paper tickets ...

You may buy the 7 Days Vienna pass in advance, but make sure to enter the correct starting date.
It comes in two flavors: (1) online for €19.70 (and not transferable) or (2) offline for €22.60 (and transferable); see details below. Online purchase requires creating an account.

(1) Use the online ticket shop, giving you the possibility to print the pdf-ticket somewhere: https://shop.wienmobil.at/en/products/T7T31_VIE

(2) You may obtain it from any ticket vending machine in subway or train stations. (The ticket vending machines in main train station are near the elevators which lead to platforms 3 and 4 at the corner to the main hall.)

Alternatively to method (1), you can install the *WienMobil' app on your smartphone, and buy the paperless ticket, which is then stored on your phone.
https://www.wienerlinien.at/web/wl-en/mobile-tickets

You just keep the ticket with you when using public transport. For the non-transferable version the name on your ID must match the name on the ticket, in case you are checked.

I understand the OBB ticket for the S7 train is separate, and is a little over 2EU each (2.10EU?), but I cannot figure out if there is a way to buy it online at OBB (just the fare from VIE to city limit).

Vienna International Airport is outside the State of Vienna. If you have already a ticket for the Vienna city area (e.g. the 7 Days Vienna pass) you need only to pay for the part of the trip which is beyond the city limits.

When you buy the ticket from/to the airport using OeBB's ticket shop, the standard ticket will show up, i.e. "adult without discounts". To the right there is a button labeled "change" for adding discounts. Click on this button and on "ADD DISCOUNT" in the next step. The list of possible discounts is very long, you need to press "show more" and scroll down for selecting "Ticket Stadtverkehr Wien (Kernzone) exkl. Jahreskarte". Press "confirm" and continue until check-out and payment.

I do see that there are REX trains and S7 trains running at that time. Can I use a REX ticket on the S7 train or vice versa?

Yes, for going to Wien Mitte train station using a standard ticket (regular or discounted) you may use any S7, REX or R train, but NOT a CAT train.

By the way, the very same ticket would be valid for going to Hauptbahnhof (main train station) using any RJ, RJX or D train.

You buy train tickets always for a certain route, not for a certain train. (The exception is heavily discounted bargain tickets for long-disance trains which are usually bound to a specific train.)