Waiting at Marco Polo

Venice airport.it has listings of all airport facilities and amenities, including shops, restaurants, transportation links.

Susan, unless something has changed recently, there is inexpensive baggage storage at the Padova train station. I think the minimum rate is for five hours. There is a nice walk or very cheap bus service into the center of town. For the middle of a weekday, I highly recommend the foods market in the center, both inside and outside, beautiful, interesting and one of the most famous in Italy.

Yes, I'd say head straight to Padova. If you land at 9 am, it will probably at least 11 am before you get to Padova.
As an alternate, you could take the No 5 bus to Piazzale Roma in Venice, store bags there and then walk around Venice for a few hours. The Padova bus stops there between the airport and Padova, so you can board it there. You could also hump the bags over the bridge to the train station and store the bags there, then take the train to Padova.

I like Sam's second alternative. Store bags in Venice and either bus or train to Padova later.

If you have the energy, i would even make it a fuller day in venice to do a late evening check-in in Padova instead of the early afternoon checkin. Have dinner in Venice first or go to Padova and have a late dinner there. You will be exhausted but then you can crash early and sleep to get over jet lag and reset your bio clock to local time.

I haven't looked at the website quoted above for facilities at Marco Polo Airport but my memory from last summer was that they were pretty sparse. There are various machines near various columns in the arrivals hall that mostly were out of order, there's the Alilaguna desk (in pretty poor shape), I don't remember any tourist information desk, there are a couple of money changers.

Maybe there's lots more but I never saw it.

I'd get to Venice or Padova.

Thanks all! Going to directly to Padova makes sense. I have read about the famous markets which we planned to use for grocery shopping while we're there. Sounds good to stash bags at Padova train station and get a sneak peek at the town center before we check in later.

I was also going to suggest going directly to Padova, but with a slightly different reason in mind. While you could stash your luggage and do a bit of touring in Venice, if you get to the point where you're ready to "crash" from jet lag, you'd still be a long way from your accommodations. At least if you're in Padova, you'll be able to check-in as soon as your apartment is ready and will have a place to use quickly if you get tired.

If you take the airport shuttle bus to Piazzale Roma, you should be able to get to the Venezia Santa Lucia station from there.

Depending on where your accommodations are located in Padova, you may be able to use the Trams from the station. As you exit the front door of the rail station, go to the right and you'll see the Tram stop which is across the street. As I recall there's a ticket booth on the way to the Tram station. There are usually Taxis waiting outside the station also, so that's another option.

If you take the airport shuttle bus to Piazzale Roma, you should be able to get to the Venezia Santa Lucia station from there.

The Padova buses go every hour from the airport. When there a couple of years ago, it was from the traffic island to the right just as you walk out the exit door. I bought the ticket at a window inside the airport, but I believe you can also pay the driver. There is a stamping machine inside the bus to validate the ticket, and a luggage bay underneath the bus to store your large bags. The bus does go to Piazzale Roma in Venice first to pick up more passengers, then proceeds on to Padova to the bus station which is right next to the train station.

As Ken says, there is the tram by the train station that goes in a north-south direction through the center of Padova. Outside the train station, there is a tourist office where you can buy weekly transit cards, 12 euro/week I believe, or a Padova Card for more money. The Padova card is either 48 hr or 72 hr. It allows entrance to many museums in town, including the Scrovegni Chapel, as well as city transit. You do need a reservation at the Scrovegni Chapel.
http://www.turismopadova.it/en/context/423
http://www.cappelladegliscrovegni.it/index.php/en/book-now/information-on-ticket-reservation

Susan,

If you need a couple of restaurant suggestions in Padova, post another note. I also found a very convenient laundromat when I was there in September.