The city of Graz has a useful page (in German) on walks in and around Graz which can be reached by public transport.
There is also a good booklet ‘ Wander-, Walking- und Fahrradguide’ which can be picked up for free at the tourist office or downloaded here with lots of good walks and hike suggestions.
I can also recommend this book (available at Moser bookstores) ‘Spaziergänge in Graz und Umgebung mit Bus, Bahn und Bim’.
For longer hikes, Styria public transport ‘Verbund Line’ has a set of brochures with good maps and transport connections which can be picked up at the transport office – Mobil Zentral, Jakoministraße 1 or downloaded here and more reachable by train here.
The Alpenverein’s office on Sackstraße 16 (opposite Kastner and Öhler) are experts on hiking trails. They have lots of hiking maps and can offer advice on which are accessible by public transport.
On weekends and public holidays you can take advantage of the Freizeit-Ticket to travel anywhere in Steiermark for €11 for the whole day.