For your visit one day should be sufficient.
Distances are mostly walkable, but getting to Postlingberg requires usage of public transport.
Worthy attractions: Mariendom; Postlingberg.
Left for another visit: Ars Electronica Center; Schlossmuseum Linz; Alter Dom.
Last visit: December 2016.
Linz is not as pretty as Graz but has a number of points of interest and plenty of colorful houses in the historic center.
Things to See
Hauptplatz ♥ is large and surrounded by interesting buildings. From here, streets in different directions offer pleasant walking routes through historic town. One direction takes you over Danube to a quieter part of town. Walk over ♥ the bridge for great views of the river and the town on both banks.
Old Cathedral was inaccessible during my visit, but Mariendom ♥♥ is very much worth exploring, impressing with its stained glass.
Postlingberg ♥ is an elevated point with Postlingbergkirche ♥ worth checking out, but the area must be more interesting in the summer. The tram line ♥ connecting it with the city center should be used at least in one direction, but probably both.
I did not walk up to the Linz Castle and left the Schlossmuseum, which is a gallery of Austrian art, for another visit. I also only walked by Ars Electronica Center, a technology museum and showcase, without visiting.