I'd recommend that you use Google Flights to scope out your air travel before you sell yourself on anything. Do a multi-city search for your potential flight dates and departure and return cities to see what comes up. Limit to nonstop or 1 stop only flights. Once you find some acceptable itineraries, set them up to be tracked so that you can be informed of price changes.
To make your final reservations you can click through from Google Flights to the airlines or go directly to them.
It's highly unlikely that your special airline is listed in Google Flights. It's highly likely that if you plan to fly from Newark to Nice, but back to the States from Athens, the cost will be VERY high because the return would be a one-way trip.
Assuming that's correct and if you are already sold on the special airline, you'd probably do better to make that a round trip Newark to Nice schedule and make either a Nice to Barcelona and Athens back to Nice flight itinerary using European airlines or if you take the train, just an Athens to Nice flight. In either case, I'd definitely spend a night in Nice on both directions because of the lack of coordination with your special airline and any others should any weirdness happen with any of them.
Out of curiosity, I used the DB Bahn website to check train options from Nice Ville to Barcelona Sants next month. The shortest time was a little over 8 hours with 2 generous train changes of 17 and 26 minutes in Marseilles and Norbonne or not so generous ones of 20 and 7 minutes in Marseilles and Nimes. Those train stations may not be the ones you want. That's still a long time, but the other options were much, much longer (like 18-21 hours and 3-4 train changes).
Having been on that train route, I can vouch for the scenery. Although the total time involved counting getting to the Nice station, waiting and boarding and the reverse in Barcelona, would total about 10 hours, the time would be worth it, especially if you've never done it before.
Below is the query link to the DB Bahn search page. You cannot buy tickets for French trains from DB, but it's a great source for seeing scheduling. The schedules for next spring won't be available for months, but you can get an idea of what's frequently available for your planning. Here's the link: https://reiseauskunft.bahn.de/bin/query.exe/en
Once you get some results, click on Show Details for basics, then on Further Information to see the stops in towns along the way. With those you can easily explore the train route on a map. Note that it will not all be beside the water.