Barcelona to Algarve Oct.2017

Have you checked on what your car-rental costs will be for picking up in Spain and dropping in Portugal, or do you plan to drive back into Spain to return the car?

In terms of a stopover point, the driving route would take you through or very near Tarragona and Valencia before you turn inland. Alternatively, you could conveniently hit Cordoba and Seville in Andalucía. By taking a somewhat longer route you could make Toledo your third base.

Between Spain's major cities the train can be much faster than a car, and the car is worse than useless inside the big cities. [Edited to remove very unfortunate typo in previous sentence.] I'd consider taking the train from Barcelona to Seville (or even flying) rather than the 10+ hours car ride (without any stops, per ViaMichelin.com). You could rent a car in Seville to see some of the smaller towns in Andalucía.

What is your actual Algarve destination?

Barcelona is one of my favorites cities. If it's your landing point from the US or Canada, your 4 nights there only gives you three full days plus your partial (and jet-lagged) first day. To me there's a lot more of interest in Barcelona than along the Algarve. But perhaps you want to spend the time playing golf or on the beach? I'm not sure what the water temperature will be like in October.

Use viamichelin.com to plan routes and see driving times and costs. For instance, the most direct way they show is Barcelona - Valencia - Cordoba - Sevilla - Faro, about 12 hours net driving time. There doesn't seem to be one stopping point. It looks like it would be better to fly to Sevilla or train to Cordoba and rent a car from Sevilla after 3-4 days in Andalucia. Then you could loop through the Algarve and return the car in Spain, to avoid the huge drop-off charges.

Chain and acraven -
Many thanks to you both for your helpful suggestions. Idea now is to make three stops, Barcelona (5 days), Seville (5 days) and Lisbon (5 days). Air or train between these cities and if we feel need, rent a car in each to further explore. This is very tentative so any ideas met with gratitude.
By the way, point of interest, we live about 5 miles away from Rick Steves offices in Edmonds, Washington! Nice to be close.
Cheers.
Steve

For flights, try www.skyscanner.com. From Sevilla to Lisbon, the best/only public ground transport is a direct bus that takes 7 hours, generally 4x/day, departing Plaza de Armas station; see www.alsa.es.

There's tons to see in Barcelona and surroundings (Montserrat and Girona being top side-trips, plus the Figueres/Cadaques area for Dali fans); you'll find that nearly all can be managed without a car, though the Dali spots require some advance researching of train and bus schedules if you don't opt for a package tour. If you do happen to be Dali fans, you might consider splitting your time and spending a couple of nights in Girona. It's a fabulous destination itself, and it puts you a good bit closer to the Dali sites. In any case, I would not want a car for five days in Catalunya.

My experience in Seville and Lisbon is not recent enough to be helpful. Edited to add: I just checked, and there's fast-train service from Seville to Cordoba. It takes less than an hour, faster than driving. At most you might want a car for part of your time in Seville. It's usually a drag to find a place to park a car in a big city, plus you'll likely pay for parking.

Barcelona (5 days), Seville (5 days) and Lisbon (5 days). Air or train between these cities and if we feel need, rent a car in each to further explore.

Go to the wiki page for the Lisbon airport. Toward the bottom there will be a chart showing all the airlines and their destinations. Another good resource is rome2rio.com (which shows flights from Sevilla to Lisbon on TAP-Portugal Airlines).

The only day trips from Sevilla that come to mind are Cordoba and Jerez. There is a lot more to see and do in and around Barcelona. Trains in Spain are at least as fast as driving, so renting a car is usually an unnecessary expense - not to mention the hassles of driving in cities and finding (and paying for) parking. You would see a lot more with a car if you wanted to string together several 1-2 night stays, but it sounds like you don't want to do that much moving around. If you stick to your 3 destinations, then I'd spend less time in Sevilla and more in Barcelona. I haven't been to Portugal, so I don't know if 5 days is a good allocation.