What would be the best mode of transportation from Barcelona to Seville to Lagos Portugal to Lisbon? We will be stopping along the way. June 2019 bus train or plane?
I would recommend the plan from Barcelona to Seville.
From there, you might consider a car. I have driven from Seville into the Algarve and it was a nice drive.
We would like to travel from Barcelona to Valencia to Malaga to Seville to Albufeira to Lisbon
What would be the best mode of transport
It is unfortunately the case that there is not great transportation between Spain and Portugal.
You will almost certainly want to eliminate car rental as an option because of the usually monstrous fee for picking up a car in one country and dropping it in another.
Taking the train from Seville to Lagos is out of the question. It takes over 24 hours as the train goes first north, then west to Lisbon, then back south.
So a bus it is. There appear to be several bus companies running that route, the largest of which is ALSA, but it appears that Eva-Bus has the most departures.
In all honesty, I do not consider the Algarve an essential part of a trip to Portugal. There are many places in both Spain and Portugal that I'd rank above Lagos. Your trip logistics would be easier if you went straight from Seville to Lisbon. There are non-stop flights under the "TAP" label but operated by White Airways, an airline about which I know nothing.