There are a lot of pleasant surprises in the Veneto and Trentino-Alto Adige regions (as well as the Friuli-Venezia Giulia region). Other than Venice (and Trent, when one can see the shroud), the entire region is way off the tourist routes -- which is a really good thing, in my mind.
My biggest suggestion is to get a Blue Guide for the region (I gave a link to it in my previous post). It gives, by far, the most detailed information of any of the major tourist guides. Using a Blue Guide, you can find something of interest in almost any small town through which you would pass.
As far as Padova is concerned, a really good time to visit it is Assumption. They have a huge party in the Prato della Valle with beautiful fireworks set to classical music for the finale. (Note: the Prato della Valle is only about 4 blocks away from il Santo)
The Shroud is in Turin (and this year would be the year to visit, as it will be on display).