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) 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 regular fare. (This is not true for the private CAT train.)