There is a pedestrian trail that goes from Ca' Vescovo (the bus stop for Asolo) up to the old town of Asolo for those that want to enjoy some beautiful natural scenery along the route. The trail head is located across the highway from the Ca' Vescovo bus stop. It's to the right about 50 metres down the road and there are signs that designate the 'centro storico' direction as well as the trail head itself. The trail takes you through the woods (dirt, rocks) and then becomes partially paved/graveled nearer the summit as you pass through an area of large homes. It ends up out on the main road into the town (follow traffic signs for 'all directions' not the parking lot!).
The trail is a bit steep and once out of the woods there is no shade, so if it's very hot or you aren't in great shape, you may want to opt for the ZEN shuttle bus to town. Also, if it's been raining, the dirt trail gets a bit muddy - leave your stilettos at home!
But your reward is being able to traipse through some spectacular olive groves and vineyards along with great views. I definitely recommend it.
You can visit Asolo on public transportation...it takes a bit of time, but it's worth it. This trip would come under the heading of 'it's about the journey AND the destination'. Across from the train station in Bassano del Grappo there is a news kiosk that sells round trip tickets to Asolo, but the stop is actually called Ca' Vescovo (look for this direction on the bus signage). It cost €4,50 round trip on the CTM line and you catch the bus a little further beyond the news stand (still in front of the train station). The buses run back and forth about every 2 hours in the summer and the last bus from Ca' Vescovo departs at 5p. The bus ride from Bassano is all uphill via a lovely, twisting, country road and takes approximately 30 minutes -- depending on how many stops they make along the route.
From Ca' Vescovo (end of line) you can either wait for the ZEN shuttle (€1 each way pay onboard) to take you up the hill (this is scheduled about every 35 minutes, but it's kind of erratic) or you can take a lovely trail to the top where the old city (Asolo Centro Storico) is located. I recommend the walk (see my post on that). And also allow time to tour around Bassano del Grappa (approx. 15 minute walk from the train to center) as it's really a lovely town too.
A few tips,
There is a parking area in the city center, near Piazza Garabaldi (with the large palace in the picture up the hill). It was full when we arrived so I decided to drive around, and around, and around, and eventually returned to park here. If it is full, I recommend just waiting as within a few minutes someone will be leaving and you can grab thier spot. The alternative is driving around the precarious, 1 way midieval streets and likely returning here.
There is another parking lot on the approach road to town (about 1 km down the hill)- Watch out as it is not open all the time, for instance on the saturday we visited, it closed at 2 p.m.
Be careful if you park on the street... from 12 until 4 p.m., you are not allowed to park in many areas.