I work as a digital product designer for a fully remote agency based in Chicago, USA. I would like to move to Austria and establish residency. I have researched the options and it is unclear how I would apply to stay past a tourist visa's duration.
My team lead lives in the Czech Republic. I would actually be reducing the time shift we currently work around.
Expat Health Insurance
If you’re thinking about taking out private health insurance, our trusted partner Cigna Global is very aware of all the difficulties that expats can face when it comes to healthcare in a new location, so they have created a range of international health insurance plans specifically designed for expats, which you can tailor exactly to the needs and ensure access to quality care for you and your family.
Sirelo has a network of more than 500 international removal companies that can move your furniture and possessions to your new home. By filling in a form, you’ll get up to 5 quotes from recommended movers. This service is free of charge and will help you select an international moving company that suits your needs and budget.
It seems your best options are a Red-White-Red Card, an EU Blue Card or a Red-White-Red Card Plus. Take a look at our visas page for Austria and the official Austrian webpage for immigration, if you haven't already.
Best of luck.