Most people who come to Austria never get to Graz, and the people of Graz are mostly fine with that. Austria's second city, with about 300,000 people, has one of the best-kept old towns in Central Europe, a UNESCO site of red roofs and Italian-style courtyards, built when the Habsburgs ruled their southern lands from here and Italian architects followed the money north. In 1529 and 1683 the Ottomans got to Vienna; Graz spent two centuries preparing for them to come here instead, and the armoury it filled is still standing, with every weapon in place.
It's also a student city, with four universities and about 60,000 students, which is why the old town is full of bars and cheap lunches rather than souvenir shops. In 2003 it was European Capital of Culture and built itself a blue blob of a museum and a steel island in the river, and it's been a UNESCO City of Design since 2011. Around it lies Styria, the green heart of Austria, with the country's best white wines, pumpkin fields and a cooking style the rest of Austria quietly envies. Graz calls itself Austria's "capital of delight", and for once the slogan is fair.
Day one is the old town and the Schlossberg, day two the cathedral quarter and the right bank, day three Eggenberg Palace and the museums, day four a trip to the South Styrian Wine Road, and day five the open-air museum at Stübing and a last evening in town. For the rest of the country, see Austria.