Train from Venice to Verona

I'd buy regional tickets at the station. All three in a day? This will definately be a "whistle stop" campaign. Padua is worth a full day on its own. Get an advance reservation for the Scrovegni Chapel.

This will be in one day. We don't plan to go to the chapel, just Verona and maybe Padua on the way back. With the regional ticket will I buy one from Verona to Padua and then another to Venice?

I can't help you with your question about tickets, but having been to all the places you mention, I will say I don't think you're going to get much out of your visit by stopping in both Vicenza and Padua on the same day. It'll be very rushed. They are both lovely and worth spending some time in. I'd choose one; probably Vicenza. In both towns the railway station is an easy walk to the main part, especially Vicenza. Verona station is also not far from the centre of town. But if you're including it in your one day, then I'd suggest you get a cab. It doesn't cost much and will save you a 15 minute walk.

Jo, it's best to do this with Regionale tickets. Just buy at the train station, validate at the little machines in the train station, hop on board and then find your own seat. Unlike the fast trains, no seat reservations permitted. If no seats, you can stand. The Regionales stop at every station. When you buy Regionale tickets in Italy, they are good for 60 days. They have no train information on them. Just Point A to Point B for 60 days. However, once you validate the ticket, it used to be good for 6 hours. It may be shorter now. The 6 hours often gave many travelers the option of "hop off/hop on" with the Regionales. Hop off, make your short visit, and hop back on a later Regionale going to point B. Travelers do this all the time at Pisa to see the leaning tower on the way to the Cinque Terre. Still, it's best to buy the individual tickets.

You can get from Padua station to the center of town using the tram in front and to the right exiting the station.

Let me get this straight. You're planning a daytrip to Verona from Venice and you want to piggyback either one or two more destinations onto it? Wow. Verona is actually worth at least two days. To give it short shrift by also cramming another destination into your day is not something I would recommend to anyone. But it's your funeral.

If you're short on time you could definitely do two of them in one day. You'll have to get an early start, do one in the morning and the other in the afternoon/evening and sure, you'll miss a lot of stuff but you can at least walk around and get a feel for each city. If I had to choose two out of the three, I'd probably leave out Padova but only because Vicenza is more compact and easier to see with a walk around the centro storico, which is also a short walk from the train station. I'm not sure I'd try to do Padova and Verona in the same day.