Padova is a city very close to Vicenza. It is close enough to drive to but I don't recommend doing it. Driving in Padova is super confusing and there are a lot of "zona traffico limitato" areas you have to watch for, or places you can't go or else you get a ticket. I took the train the other day and it was only a few euro and took less than 20 minutes to get there.
Padova is famous for their food market. Every morning you can find fresh foods being sold in the piazza della frutta. There is also a meat market under the arcade of the building you see here.
"At 90.000 square meters, "prato delle valle" is the biggest square in Europe and probably one of the most beautiful in the World. Historically a Roman theater and later a fairground, it was redone in 1775 to the present layout: a large central grassy area, surrounded by a statue-lined canal, then a broad expanse of flagstones before a couple lanes of traffic are allowed to trickle around it in the distance. Saturdays the square hosts a giant market." (http://wikitravel.org/en/Padova)
Next time I go to Padova I would like to reserve tickets in advance to see the amazing frescoes in the Scrovegni chapel.