Yes, ÖBB as well as DB and SBB offer various discounts for families.
With ÖBB , small children from 0 to 5 years always travel free of charge when accompanied by an adult. With the Vorteilscard-Family, up to 4 children under the age of 15 travel free of charge for only € 19 per year. More information here.
DB is also family-friendly: in many cases, accompanied children up to and including the age of 14 travel free of charge. Discounted family reservations, DB Family Card, family compartments and infant compartments can be reserved. More information here.
SBB: Both accompanied and unaccompanied children under the age of 6 travel free of charge. Children travelling alone or accompanied from their 6th to their 16th birthday pay half price or the minimum fare for tickets. More information here.
At the Verkehrsverbund Tirol (VVT), children up to the age of 15 can travel free of charge on school-free days with a Climate Ticket and EurogioFamilyPass on trains and buses.
Many holiday regions in Tyrol offer free use of local public transport with the guest card .