The lower town (Città bassa), although prosperous and by no means ugly doesn't have a great deal to interest to the sightseer, but the old town (Città alta) sat upon the hill contains a wealth of ancient buildings, and some mighty city walls, as well as stunning views across the flat Lombary plains to the south. Don't try to drive into the old town, instead park at one of the many underground parking garages in the lower town, and then walk to the funicular railway station. It's possible to walk, but the climb is steep, and the train is interesting intresting in it's own right.
Take the funicular up the hill. Try to get a seat at the lower end of the train, so you can enjoy the fabulous views to the south from the train. Follow the main street to reach the Piazza Vecchia and explore the many historical buildings on all sides, paying particular attention to the Basilica di Santa Maria Maggiore and the Duomo. There's also an art gallery and other stuff to see. The fountain and square is a beatiful place to see. Then carry along the main street to a small square. Turn right and exit the city walls onto a terrace with views into the foothills of the alps. Savour the shade of the small park, before heading back into the city walls, and carry on back to the main street.
Then thread your way through the backstreets before arriving back on the main street near the funicular station, and then had up the steep street to the Roca, or Citadella. Climb to the castle and admire the views east, in gardens full of old field guns, and memorials to the fallen.
Linger in the cool shade of the park, befor returning heading up the castle to see more views over the city, before returning to the funicular station. if time allows you can walk down a series of steps and steep streets to the bottom station of the funicular.