how to arrange this travel?

Lisbon is not easily accessed from Spain by land. It is best to fly. Seems odd but that is the way it is.
Perhaps by bus too.

I have not travelled to Spain nor Portugual. However, I have had several trips across the pond from the US. This sounds like an extremely ambitious, if not unrealistic plan.

Am I understanding correctly that the 2 days in Barcelona are part of a total 6 days (Is this 5 nights?) you have between the cruises?

Do you leave Barcelona for Athens? Is that within the 6 days? What time of day? I think it makes sense to be in your city of departure the night before your cruise leaves. (disclosure: I have never been on a cruise, but it would be a shame if transportation issues had you stuck in another city as your flight took off.

What do you hope to do in each city? A city tour, meal, sleep and then move on to the next city? It seems to me that you only have one night for each city. Transit to and from will eat up a lot of your 6 days.

If I had the limited time I believe you have, I would take a train to Madrid or Valencia and enjoy a few days there before returning to Barcelona (or leaving that city) for flight to Athens.

Another idea: I see it takes almost 4 hours by transit or car from Barcelona to Valencia. Without study, I don't know the towns between the two cities, but I imagine it would be nice to visit towns and villages along the coast en route. Rent a car in Barcelona. Drive to Valencia, visiting (staying?) in a villages along the way.

I would not want to insert an extra flight into this itinerary, presumably you will already be flying to Athens.
Barcelona and Valencia would be plenty with 6 days, and you could even drive between them if you wanted to see some smaller places/scenery. But for the cities themselves, train service would be the way to go.
You can fly to Athens nonstop form BCN, not from Valencia, so you could return to BCN for your onward travel and store your large luggage and just take a smaller bag to Valencia. I would save Lisbon for sure.

Do you have to fly to Athens from Barcelona?

A one way car rental is going to have an ugly drop fee. Driving r/t from & back to Barcelona is at least 1600 miles in your 6 days. Lots of driving and not so much "being there" . Train service between Spain-Portugal is limited and cumbersome.

It might make for a more enjoyable trip to do Barcelona, Valencia and Madrid ( use trains) and fly to Athens from Madrid

My $0.02.

Valadelphia, I agree.

What Joe said!!!

6 full days in Barcelona with also Valencia, Madrid, and Lisbon. That is a full plate. Too full. That is 1 1/2 days each place. Plus the travel between. Even dropping Lisbon leaves you two days each of three cities, but there is the travel between. I sense a big FOMO here. Plus seems you are cruisers. Meaning that your usual is probably just single days in a port. Those six days probably seem to you to be a whole lot of time. It’s not.
Approx train times between Barcelona Madrid is 2+ hours. Same between Madrid and Valencia. Then Valencia back to Barcelona is 3 hours. You need to do some research on your own with your ideas in mind. The above is a triangle.
You would need travel time and finding your hotel, checking in, dropping your bag and wandering until check in time. You probably are not used to that mode as opposed to cruising. You will be inefficient. Your time in each place will be only one full day/two nights.
FOMO is rampant on these forums as you can see by so many proposed itineraries and the help they ask for. You can go many places in a certain timeframe. It is possible, but it is a flash deal. Then again, your style may be OK with that.

6 days, forget traveling to Lisbon. Even going to Madrid or Valencia will take up lots of time, especially driving a car.

You only have five full days since you would need to return to Barcelona the day before your next cruise.

You could easily spend those days in Barcelona, then visiting Monseratt and Girona.

Train from Barcelona to Valencia is about 3 hours; to Madrid 2.5 hours. They're easy. Don't understand the naysaying. Lisbon, though, is another matter.

Do you have days in Athens after your cruise ? If not, then split the 6 days 3 each in Barcelona and Athens. If you do already have Athens days scheduled, choose 1 of the other cities and split evenly with Barcelona. I would choose Madrid

I want to thank everyone for your valuable comments and suggestions!!

We are going to scale back the plan and visit only two cities, Barcelona and Madrid in the 6 days. We are going to visit Athens after our second cruise (which starts and ends in Athens), which will give us plenty of time.
Again, thanks for all the advices you guys have given!

Forget Portugal - but consider Seville and Granada - you can get there in a day by high speed train. Also Girona is short day trip from Barcelona

Good Plan B! Barcelona and Madrid, fly to Athens from there, spend more time in Athens after the second cruise. The train from Barcelona to Madrid is easy and fast. Plenty to do in both cities for the time you have. Assume you will return to Spain for the rest.

Just to wrap my head around the question...it goes like this:

1. Florida -> Barcelona transatlantic cruise;
2. Barcelona for six days;
3. Several (many, I hope) more days each for Valencia, Madrid, and Lisbon;
4. Flight (hopefully) from BCN to Athens for next cruise.

I hope #3 is accurate. IOW you have a nice chunk of time for (at least) Madrid and Lisbon. Valencia is beautiful and lovely to visit, but Madrid and Lisbon - to me - require several days each especially if making the effort to travel there from Barcelona.

If I was wrong about there being a large gap of days between arriving at BCN and needing to be in Athens, I would absolutely not try to squeeze that itinerary in. When we flew to BCN last year, we flew TAP which was routed through Lisbon. I'd consider that - cruise -> BCN (several days) -> fly to LIS (several days) -> fly to Athens -> cruise -> fly home. Maybe an open jaws with multiple legs.