Im flying to Zurich

Are you returning the car to Switzerland? Dropping off a car in a different country may incur significant extra charges. You may consider a train and short bus ride option. I use Rome2Rio for looking at transport options. Here is what I turned up: https://www.rome2rio.com/map/Zurich-Airport-ZRH/Dornbirn

Safe to drive? My goodness, yes! You could also take a train from the airport to within 3/4 of a mile of your destination in 2 hours.

If you do rent a car, you need an Austrian vignette for your car entering Austria if you drive on an Austrian motorway. Since the Swiss autobahn runs parallel to the border in that area, you can probably avoid driving on a motorway, unless you decide to do some touring into the mountains. A car rented in Switzerland will come with a Swiss vignette already attached, so no worries on the drive there.

Edit - Come to think of it, I believe Austria requires an International Drivers Permit to go with you regular drivers license. Pick one up at AAA.

And yes, Zurich airport is the closest International airport to western Austria. Everybody in those parts uses it.

No Maria, we are not returning to Zurich. Thank you very much for the link!!!! :)

@Sam, thank you very much for your quick response!!!!! it help me a LOT!!!! Have a Magical Day!!!

That address is "Messeplatz 1". Messeplatz = Exhibition centre square, obvious address for an Art exposition.

Yes, you can rent a car and drive. You would need to return the car back to Zürich airport (or elsewhere in Switzerland) at the end or pay a large surcharge.
Absolutely no issue driving across borders. There is no passport check, customs only; you probably won't even get stopped.
two minor issues:
1) You need to buy an Austrian Vignette to pay for tolls.
2) Depending on which country issued your licence, you may need to get an IDP (International Driving Permit) to certify that licence as valid.

However, if I was going from Zürich airport to Dornbirn, I would take the train. a lot less stress.
You can look up trains on the SBB (Swiss Federal Railways) website: https://www.sbb.ch/en
"Zürich Flughafen" (airport) to Dornbirn takes 2 hours, departure every hour and costs CHF 42.

Edit: I just noticed your post "we are not returning to Zurich.". Where are you returning to? See above comment about needing to return any rented car in Switzerland.

I found the website for "Messe Dornbirn" (Dornbirn exhibition centre).
There is an exhibition "Art Bodensee" 26th - 30th June. I guess that's it.
it has a webpage in English: https://artbodensee.messedornbirn.at/art-bodensee-contemporary-art-fair/

Thank you very much Chris!!!!! 😊

Can anyone help me how to get from the airport to the train station? I can find the address for the train station in Zurich.
Please

Train Station is downstairs in the Zurich Airport.

Yes, the train station is across a pedestrian bridge to a parking/shopping structure, then escalators down to the basement. Just follow the train icons.
https://www.google.com/maps/@47.4500557,8.5632622,3a,37.5y,42.3h,103.65t/data=!3m8!1e1!3m6!1sAF1QipMCspCp60jeNCMAi6XaTvRkQBvE9cNVbQqqlc6b!2e10!3e11!6shttps:%2F%2Flh5.googleusercontent.com%2Fp%2FAF1QipMCspCp60jeNCMAi6XaTvRkQBvE9cNVbQqqlc6b%3Dw203-h100-k-no-pi-0-ya54.97501-ro-0-fo100!7i10240!8i5120

Can anyone help me how to get from the airport to the train station? I can find the address for the train station in Zurich.

There is a train station at Zürich airport. It is called "Zürich Flughafen" ("Flughafen" is German for airport). It is signposted from arrivals, less than 100 metres walk. You do not want to go to the train station in Zürich city (called "Zürich HB").

You can look up trains on the SBB (Swiss Federal Railways) website: https://www.sbb.ch/en/
Enter Zürich Flughafen to Dornbirn, and a date and time.

The Zurich airport train station has a staffed ticket counter. Ticket agents are very helpful. I don't know if credit cards without PINs work in the ticket machines, but the ticket service staff took signature credit cards.

Thank you very much everyone!!!!! This page is amaizing!!!! 👏🏼👏🏼👏🏼

The kiosks for train tickets did seem to require true chip and pin cards but I managed to get around this by using Apple Pay to make the purchase at the kiosk instead of having to use the staffed counter.