Rental car in Lindau Germany

At which station does the train arrive in Lindau?

Whichever station you wish to use - Lindau Insel or Lindau Reutin - it's your decision, one you would base on the location of your rental outlet, perhaps. Or are you staying in / visiting Lindau as well? If so, Lindau Insel (island) is likely your destination.

You will actually need to use 3 different trains for the journey to Lindau Insel. Getting to Lindau Reutin requires either 2 or 3, depending on your travel hour.

I can't advise you on car rental outlets. That's another very messy decision to make. It seems possible that the best rental company for YOUR needs, budget, etc. might not be in Lindau at all but perhaps in some neighboring town.

FWIW, I would consider instead the train in Germany. Passes like the Bayern Ticket and the Deutschlandticket will get you to your destinations easily enough and cheaply as well.

Where is a good place to rent a car in Lindau that is in close proximity to that train station.

The only rental company where this applies is Hertz, behind the Lindau-Reutin station (where the train from Switzerland stops), approx. 500 m away. Europcar is 1.5 km away, Sixt over 2 km. But you could either take a cab or have the rental car come to the station. The easiest way is to leave the train at the Lindau-Reutin station, then you don't have to change trains (the EC train from Zürich doesn't stop at Lindau-Insel) and it's also easier to pick up the car there than at the station on the island in the middle of the narrow old town.

Take some time afterwards to explore Lindau. There is a parking garage directly on the island. Drive over the bridge and then keep right at the city wall; it's signposted and you can't miss it.

You will need an International Drivers Permit to drive in Austria and Germany.

Thank you for the advice about car rental places and especially about needing an international driver's license!