Private secondary schools are a little further afield, but many offer private bus services to pick up and drop off in Riverhills, including Forest Lake College (Forest Lake), West Moreton Anglican College (Karrabin, Ipswich), Westside Christian College (Goodna), St Peters Lutheran College (Springfield Campus has a private bus service). Other nearby private schools with good public transport access include St Peters Lutheran College (Indooroopilly campus), St Aidan's (girls, at Corinda), BBC (boys, at Toowong) and St Brigidine's (girls, Indooroopilly).
The closest private primary schools are the Good News Lutheran School at Middle Park (3 minutes drive from Riverhills); and the Our Lady of the Sacred Heart Catholic School at Darra (8-10 minutes drive from Riverhills). My elder daughter attended GNLS from Prep to Year 7, and I will be sending my younger daughter there too ... highly recommended! The Catholic school is also very popular with locals.