From a practical point of view, you won't need any documents as nobody checks anything. I've travelled extensively between countries in the Schengen zone and not once have I had to produce my passport at a border when coming by train or car (one exception during a lockdown and me travelling on an emergency). Even on plane travel it is rare.
However, the law in several countries requires non-citizens to have a passport with them at all times. In Austria, if you are a EU citizen, you have to have a passport with you. If you are a non-EU citizen, you have to have your Aufenthaltskarte with you. (see here or here).
So even if you don't cross the border, you need to have your card on your person. I don't know the laws in Italy or Slovenia. Just make it a habit to take your card, because why not?
(of course, your bus company may demand that you have a passport, or a red hat and a teddy bear - as long as they're not a public transport and clearly post them, they can make whatever rules they want)
Update: According to Mark's comment below, the information about having to carry an ID document is apparently wrong for Austria, despite multiple online sources saying so - see here for an official government page. You might still want to check what the law in Italy and Slovenia says.
Update2: An important comment was moved to chat: In Slovenia, you apparently do need to have an ID card.