Would you know what does VE S.L. MESTRE mean when searching for trains?

VE S.L. stands for Venezia Santa Lucia. The train starts there. The red "Mestre" underneath just indicates that it stops there so people can pick it up there as well. This is a nice ES train (Frecciargento) that goes from Venice to Rome with no train changes. It does stop a few times along the way.

Sonia, Lola beat me to it! Mestre is a major hub station on the mainland; the schedule shows that the trains do or don't end (or begin) at Mestre. Even with a carry-on, it's hard to swim the Lagoon ;-)

If, in the future, you find a train that really fits your schedule the best but it doesn't go on to Mestre, there will be a later train that you can take...

So, the answer to your question is, it's not either/or, it's both!

"want a direct train to Rome non-stop!"

As a friend of mine used to say, "People in hell want ice water". Or, to quote the Rolling Stones, you can't always get what you waaant."

This just isn't how European rail works. The direct (i.e., no change) trains from Venezia S.L. to Rome stop in Mestre, Padova, Bologna, and Florence.

You can get non-stop b/n Rome and Florence, but it's a long walk to Venice from there...

You CAN get trains with no connections - I assume THAT'S what you're really after.

Sonia, you will probably find that every train departing Venezia Santa Lucia will stop at Mestre. It may not show in red in the train schedule but if you go further into the detailed section, you will see Mestre and Padova as the first two stops on the high-speeds.