Kids under 6 are free on almost all public transport in Austria.

For older kids there are a lot of discounts, but many of them are only cheap if you use public transport regularly. (eg. rail card family from the ÖBB for 19€ a year, or upper austria regional tickets for school children for 77€ a year, etc.)

One problem is that it's not just train tickets, but you also need tickets for bus/tram/subway, and every city/region has different systems, so for every trip I spend about 15min trying to figure out which tickets to buy.