Austria has a very good public healthcare system, and your EHIC cards allow you to use it. Most people don't feel the need for private health insurance, but if you need more than just basic coverage, private health insurance is an option. Have a read of our page on healthcare in Austria for some more information on their public healthcare.
Are you moving to Austria permanently?
Hope this helped,
Once you're in Austria, you have to register to get access to the public healthcare system. After that, you'll get a green e-card that processes all claims and such. The process explained in more detail in our guide to healthcare in Austria, so have a look there.
How long have you been in Austria?