If I will be going from Switzerland to Austria by car, will I need a passport?
A passport, not necessarily. Your national ID card also work and beyond that, the European Union maintains spreadsheets of recognised travel documents and one of those you are very strongly recommended to keep with you. There are three spreadsheets:
The short answer is: yes.
The long answer: Switzerland and Austria are both part of the Schengen area, where border controls are generally reduced or non-existent. Nevertheless, while it's very possible that you might cross the border without any official inspection taking place, you should always carry a passport (or equivalent identity document) when travelling internationally. Spot-checks are possible even at borders within the Schengen area, and even ignoring that possibility there are many countries (including Austria) which require you to have ID available at all times.
Depending on your citizenship, a national ID card might be a valid substitute for your passport. But you should carry some form of government-issued document which can be used to confirm both your identity and your legal presence in the country.
If you're a citizen from a country in the Schengen area you need to have your national ID in case you get stopped and your identity needs to be confirmed. If you're not you do need your passport.
Most of the time you will not be checked as there are no regular border controls within the Schengen area. If you get checked as a citizen of an EU/Schengen country you will need to somehow prove your citizenship. This doesn't necessarily mean a valid travel document but in the end, it's just asking for trouble not carrying your national ID.