Venice airport to Padova

Only if you want to waste time and money. It's not a vaporetto, but an alilaguna boat and it costs 15 euro and takes 42 minutes. It starts with a 300 yd walk to the dock from the airport to catch the boat ("A" or Arancio line), it drops you at Guglie dock which is another 300 yds from the train station with a stepped bridge to cross with your luggage.
Or take the hourly Busitalia bus from the airport to Padua train station. The bus leaves from right outside the airport exit. You can buy the tickets at the window inside the terminal (not sure, but less than 10 euro I think) and the trip takes an hour and 5 minutes. Your luggage is stored underneath the bus.
Edit- I paid 8 euro in May 2012.
You can also get a bus to Mestre station and take the train from there. The train fare is 2.90 euro and takes anywhere from 15 to 36 minutes depending on the train. They are about every 15 minutes or so. The bus is 6 euro and goes every half hour and takes 26 minutes.

Thanks so much for the detailed information.

What's usually described as wonderful from the airport into Venice is neither a vaporetto nor Alilaguna, but a private water taxi. Picture jetting across the lagoon as Venice gets larger and larger in front of you. (The Alilaguna, on the other hand, is very loud, not all that fast and you sit so down low in the water that you don't really get to see much.) However private water taxis are ferociously expensive at 80 to 125 euros. Take the bus as noted above.