Venice - how to get from Santa Lucia train station to Mestre station and cost?

Not to worry - there's a train from the Santa Lucia station to the Mestre station every 10 minutes. It's a Regionale train, so very inexpensive and reservations not needed.

It can easily be taken care of.

First, is your train coming in from Trieste, so that you are sure that it doesn't start in Venezia Santa Lucia? Most trains which originate at Santa Lucia also stop at Mestre, so perhaps it was just a ticketing oopsie.

Can you share which train it is?

If, indeed, you can't get your train from Santa Lucia, buy a Regionale ticket (very cheap) to Mestre (careful, there are a couple of stops between that the most milk run trains stop at - I've got off the wrong station once) and just add enough time that you will have a comfortable time to change platforms and trains in Mestre. No escalators or lifts at Mestre, at least none that I have ever found, so be ready for the stairs.

The cost? 1.25 EUR. Equivalent to 1/6 of a vaporetto ride.

The OP says that the actual train originates in Trieste and does not go through Venezia Santa Lucia. Rick Steves Travel Forum guide lines prevent us from advising people to break the law, i.e. riding without a ticket. Being as the ticket from Venezia Santa Lucia to Venezia Mestre is only 1.25 EUR, there is no reason not to buy a ticket. In the remote occasion that they will check tickets on the train, the fine would be way more than the ticket price.

In case it would be more convenient for you, you could alternatively get to the Mestre train station on a bus from the Piazzale Roma - just inquire at the ticket booth.