Barcelona to Arles via train

The trip from Barcelona to Arles will start out on a fast TGV or AVE train. At some point you will need to change to a TER (French regional train). There are several different places where that could happen: I see Narbonne, Montpelier and Nimes, and I only checked a few hours' worth of departures. Travel time to Arles will depend not only on how much of the trip is on the TGV or AVE but also on how long you must wait at the transfer point for the regional train. So you'll want to take a careful look at the schedules. For checking out just the timing (not the fares) I find the Deutsche Bahn website the easiest to use.

TGVs and AVEs require seat reservations; I think the international reservation fee on the TGV could be rather significant. To compare rail-pass cost to ticket cost you will need to look beyond the Deutsche Bahn website. The SNCF website is here: https://en.oui.sncf/en/. Check your planned travel date to see what kind of deal you could get if you buy now, then check the fare for the same day this week to see what it might cost to purchase much closer to your travel date. Because of a varying mix of fast-train mileage and regional-train mileage and varying levels of demand, you'll probably find some price variation for different departure times.

For the Arles to Paris travel leg, you'll again have a mixture of regional train and fast train (TGV). I see that the transfer can be made at Nimes, Marseille, Avignon and Lyon. I'd avoid Avignon because you'd also need to switch stations there. The Lyon station is modern but extremely hectic and crowded at some times, so it wouldn't be my top choice, either. I haven't been to Nimes but assume that station is a good bit smaller than Marseille-St. Charles, so I'd try for a departure with the regional-to-TGV transfer being made at Nimes--unless someone with actual experience at Nimes says there's an issue there.

There should be first-class cars on all the AVEs and TGVs. I don't know whether you'll find that class of service on any of the regional trains. If you click on "Show details" on the Deutsche Bahn website, you'll see a "2nd class only" statement if it applies.

Regarding the Eurail pass: there is no way it will be worth the price given that you only hava two journeys. And they do not give you flexibility because of mandatory reservations.
So, look at oui.sncf as mentioned above, and book now for the best prices. Barcelona to Arles costs about 70-90 euros in first class on a random date in November. You will need to transfer along the way, probably in Nîmes or sometimes in Narbonne. There is no first class on the second train you need to take to Arles but the regional rolling stock is quite comfortable regardless.

For Arles to Paris, the bus to Avignon TGV was sadly discontinued - so you have a choice between some direct trains at possibly inconvenient hours (early morning or late afternoon), trains with a transfer in Nîmes (best choice in my opinion), or in Avignon Centre then Avignon TGV (but here you need two transfers...), or in Marseille (which makes you backtrack!).
Or a taxi from Arles to Avignon TGV will cost about 80 euros.

I'm traveling from Barcelona to Arles in a couple of weeks. You've gotten great advice about trains from acraven but I'd like to add that we will be changing trains in Nimes. I've arranged to get there around noon and then we'll leave off our bags at l'Hotel d'Abalone, avenue Feucheres, at right when exiting the train station (5 euros/bag). We'll spend about 5 hours in Nimes seeing the arena and the Maison Carre before getting on the regional train which will take about half an hour to get us into Arles.

I've been to both Barcelona and Arles before - they're great and you'll have a terrific time!! (Nimes will be new for me and I'm looking forward to it!)