Train from Lyon to Barcelona with a stop at Girona. Possible to stop at Girona/Figueres?

Hopping off and on (in that order) won't be possible with a ticket from Lyon to Barcelona, being valid for that specific train.
The SNCF, being the French railway company, won't sell tickets for internal travelling outside France. With the SNCF you can buy a ticket from Lyon to Girona. Girona to Barcelona must be bought with the Renfe, the Spanish railway company. Best done at Girona station.
Having said that, I have serious doubts whether it's wise to do Girona en route to Barcelona, if only for the hassle with your luggage. I'd would do it as a day trip from Barcelona.

www.seat61.com suggests several alternatives to buying tickets through the Spanish train operator. I used www.trainline.eu for part of the route you are suggesting with pleasant results. But I agree that your stops will require separate train tickets.
Girona is a scenic spot with ambitious restaurants and worth a full day's visit (which is to say two nights.) Its modern station has escalators from the train platforms to the lobby to help with luggage. The real challenge is dealing with bags on the train itself. It may be a double-decker so you could be hauling your bag (bags?) to the upper level. Packing light will be doing yourself a favour.
Figueres' main attraction is the Salvador Dali museum where line-ups to get in can force a substantial wait. Worth it, in my opinion, to see the old provocateur in action. I didn't use the train station and don't know what you might do with your luggage during a brief stop.

Lyon to Barcelona would be best on a TGV and then a day trip to Girona. Otherwise you'd have to switch to a local train somewhere which could make the trip very slow.

Hi everyone thanks! I think I may head straight to Barcelona to dump my luggage and do a day trip to Girona. Any recommendations for Girona or tips for attending the Sant Narcís Fair and Festival? Thanks so much!

There is a great deal to enjoy in Girona, even on non-festival days. It's also a great base for visiting the Dali sights in Figueres and outside Cadaques. I definitely wouldn't want to have just a couple of hours there between trains.