Fast train Venice to Padua and Verona

Venice to Padua is via very frequent trains. Take a Regionale Veloce, just as fast as a Freccia train at a fraction of the price. Just buy a ticket at the station and validate the ticket before boarding. Same deal Padua to Verona. You can take a fast Freccia train from Verona to Milan, but again, the Regionale Veloce is almost as fast and cheaper spur of the moment.
To see schedule and prices go to http://www.trenitalia.com/tcom-en
Use Italian station names, Venezia S. Lucia, Padova, Verona Porta Nuova, Milano Centrale.

Trenitalia is the national railroad system. And most of the time you don't have a choice. It them or walk.

Regionale Veloce from Venezia S. Lucia to Padova takes 26 minutes €4.25.
Padova to Verona Porta Nuova takes 1 hour on a Regionale Veloce €7.30, on the Frecciarossa, 42 minutes, €18.90.
Verona to Milano Centrale 1 hour 50 minutes on a Regionale for €12.75 or the Frecciarossa 1 hour 20 minutes for €20.90.

Download the TrenItalia app or use their website to see complete schedules.

You sound like a worried 1st timer. It's really not that big of a deal. Go to the train station and buy a ticket either from the self service kiosk or the ticket counter for your date of travel. Millions of people (including visitors) do it regularly. You'll be fine.

Go to the train station and buy a ticket either from the self service kiosk or the ticket counter for your date of travel. ...

...then, on the day of travel remember to time-stamp the Regionale trains tickets before getting on.

Time-stamping may be the main difference between italian trains and those you have already taken in your home country, but It can be an expensive difference so don't forget it.

Thank you so much for all of the wonderful help.
It sounds so much easier than I was imagining.