Spanish domestic trains reservations are a pain to get (they don't like Interrail/Eurail passes for some reason...).

The TGV between Paris and Barcelona should be available to book (or will soon be) but be aware that there are expensive seat reservation fees (35€) for Eurail passes. If that is the only train you are taking, please have a look at regular tickets on sncf-connect.com instead of buying a pass.

I do not understand the rest of your message. Barcelona - Mallorca is either done on a boat or on a plane yes. But how do you want to reach Sevilla ? The easiest way would be to catch a direct flight between Mallorca and Sevilla.

To clarify….. I know Mallorca is done by boat or plane. We are flying from Mallorca to Barcelona then taking a train down to Seville. We want to do that all in one day. So in order to book the flight from Mallorca to Barcelona, I need to know when the train leaves Barcelona so I can time it right. We don't want to fly to Seville directly because we want to experience the countryside by train. 

What is the high speed train in Spain? I've now learned that Eurail is not a train, but a pass for all the European trains and each country has its own train system. Do you have to buy a Eurail pass or can you book directly with a train?

Ok I didn't understand that you want to fly to Barcelona return.

Just a note : the high-speed line between Barcelona and Sevilla runs through the middle of the country and you won't see the sea.

Right now the direct AVE train leaves Barcelona-Sants station at 3:15pm and arrives in Sevilla at 9:15pm. We do not know yet this summer's schedule but it should be similar.

Leave enough time to go to Sants station from Barcelona Airport.

There are more connections but with a change in Madrid.

To buy this train use its operator's website renfe.com (I don't think tickets are available yet for this summer). Sometimes it doesn't accept foreign credit cards, see advice here : https://www.seat61.com/train-travel-in-spain.htm#How_to_buy_train_tickets_for_Spain

To buy the TGV high-speed train between Paris-Gare de Lyon and Barcelona-Sants, use sncf-connect.com

Don't buy a Eurail pass (valid on most European trains), it won't be worth it + you need expensive seat reservations.

Thank you. That is helpful. So Renfe is in Spain and TGV is in France? Then what is AVE? 

AVE is the name for Renfe’s high-speed trains (it means Alta Velocidad - high speed). TGV is the French equivalent - “Train a Grand Vitesse”, and SNCF is the national operator in France.

You might find this guide to train travel in Spain (and other countries!) useful.

https://www.seat61.com/train-travel-in-spain.htm

Booking a ticket using Eurail website is a simple process. Follow these steps:

1. Go to the Eurail website at www.eurail.com and select your home country and the country you will be starting your journey from.

2. Choose your travel dates, the number of travelers, and the type of pass you want.

3. Select the train routes you want to take and the classes you want to travel in. You can also choose any additional reservations you may need.

4. Review your itinerary and make any changes necessary.

5. Add any extras you may need, such as seat reservations or luggage storage.

6. Enter your personal and payment information.

7. Review and confirm your booking.

8. Once your booking is confirmed, you will receive an email with your ticket and reservation details.

@Theodore seat reservations for most countries work on eurail.com but not all. Reservations in Spain are impossible to get there.

And there are booking fees on eurail.com for every train + they send paper reservations by mail for example for Norway.

ÖBB has no booking fee + sends the reservation by email