How to book a train ticket round trip from La Spezio Centrale to Montorosso.

You do not need to pre-book a ticket. It is a regional train, and they run all day long. Just go to La Spezia and buy it. Don't forget to validate it!
You can get a Cinque Terre card at any of the Cinque Terre towns at the train station ( I believe). I always get mine in Vernazza.

Upon arrival at the La Spezia train station, you can easily buy local train tickets or buy a Cinque Terre pass that includes train transportation. There was a recent post that had good information about pre-purchasing a Cinque Terre pass, but unfortunately I can't locate it. Hopefully someone else will remember and pass the information along.

There are no round trip tickets. You buy a ticket to and a ticket from. You can buy both at a ticket machine or ticket window. Since these are Regionale train ticket do NOT buy on line. On-line tickets come pre-validated with a short time window. Buy at station and validate just prior to boarding. Then the ticket is good for four hours. Do not validate the return ticket until ready to return.

rubinka, buy a ticket when you get to La Spezia. If you are going during high or shoulder seasons, it'll cost you 4 euros. La Spezia to Monterosso is only 19 to 23. minutes. Be sure and validate it before you get on the train!

Before talking about the CT card, how long are you going to be in the CT? There are two different versions of the card: one for just trekking and one for trekking and trains.

http://www.parconazionale5terre.it/Ecinque-terre-card.php

Red trail? Are you sure you don't mean Blue Trail (Sentiero Azzurro)? Not all of it is currently open but there are a couple segments which are.

We have one day, my plan is by 9 am La Spezia to Riomaggiore, then up to Montorossa, south to Vernazza, Corniglia, dinner at Billy’s in Manarola, back to La Spezia to drive back to hotel just southeast of Florence. I read that one trail is better but now not sure if it is as Red Ave Blue give versus. I’m hoping to get in a little hiking as it’s museums and driving for 8 days.

rubinka,

Could you clarify which trails you're planning to hike? Your reference to "Red Ave Blue give versus" was somewhat confusing. The most popular trails are the Sentiero Azzurro (Blue Path) trails but only the Monterosso - Vernazza and Vernazza - Corniglia segments are open. There are other trails that run higher in the hills but they're a more robust hike.

We have one day, my plan is by 9 am La Spezia to Riomaggiore, then up
to Montorossa, south to Vernazza, Corniglia, dinner at Billy’s in
Manarola, back to La Spezia to drive back to hotel just southeast of
Florence

OK, if you want to do any of the Blue Trail (SVA2 on the map, and what I think you're talking about) I'd skip Riomaggiore, train directly to Monterosso and hike the Blue from there to Vernazza (SEE BELOW) and onward to Corniglia. Then get on a train to Manarola for dinner and another onward to La Spezia.

This plan is only viable in dry weather. Skip the hike if it's raining (the trail will likely be closed anyway).
The Monterosso>Vernazza section is showing as closed at the moment so you may have to skip it unless it has re-opened by the time you visit
http://www.parconazionale5terre.it/Esentieri-outdoor.php
http://www.parconazionale5terre.it/Eiti_dettaglio.php?id_iti=3578

Vernazza to Corniglia:
http://www.parconazionale5terre.it/Eiti_dettaglio.php?id_iti=3577

You will need a CT Trekking card to hike any segment of the Blue.

Back to your original question re how to book your tickets: If you have a smart phone you might consider downloading the Trenitalia App. We found it to be quite helpful on our own recent trip. With it you can research and book your tickets electronically with your credit card- no standing in lines at the ticket windows or kiosks. The tickets will be sent to you as attachments on a confirmatory email, and will already be time stamped which eliminates the need to validate them prior to boarding.
And if you're travelling as a couple or a group only one person needs to establish an account - you can book as many tickets for as many people as you like by simply foillowing the cues on the booking app. It really is quite convenient. With it I was able to book our tickets while sitting in a cafe sipping a cappuccino rather than having to brave the mosh pit at the station trying to buy a paper ticket. Recommend it highly.

this link may be helpful https://www.seat61.com/Italy-trains.htm