Graz is small enough to walk end to end, so the real choice is the old town's stone lanes, the scruffier Lend side across the Mur, or the cheaper streets around the station.
Innere Stadt — the red-roofed core
Everything inside the bend of the Mur: Hauptplatz, Herrengasse, the armoury and the stairs up the Schlossberg, all within ten minutes on foot. You pay the most here, and the trams below the window start early.
Lend and Gries — across the river
The west bank around Lendplatz and the Kunsthaus: Balkan and Turkish grocers, cheaper rooms, more paint on the walls. Ten minutes from Hauptplatz over the Hauptbrücke, and the side of town that stays open when the old town shuts.
Geidorf — the university side
North of the centre behind the Stadtpark, mostly flats and student houses, running up towards the Karl-Franzens-Universität and the Hilmteich pond. Quiet at night, on the tram network, and usually cheaper than the core.
Around the Hauptbahnhof — practical tip
Annenstraße runs from the station to the old town in a straight 20-minute walk or a few tram stops, and beds near the station are the cheapest in the city. Worth it if you arrive late or leave early; the street itself is traffic and chain shops.