Portugal itinerary questions - car rental and must do's

Welcome to the Forum. Your inquiries are quite general so I will respond as such. I do not have experience with car rental in Europe so cannot help you there. What made you interested in Portugal? Perhaps start there - your interests, activities, food/restaurant preferences and style.

Have you looked at any travel guidebooks or videos for Portugal to get a sense of what the country has to offer? Rick Steves has a very good guidebook and also has quite good videos for the country. Explore this site a bit to get a feel for what interests you and plan from there. If you’ve never traveled to Europe, Rick’s Europe Through the Back Door book will really give you an understanding of the logistics of travel all over Europe. Of course, there are other resources for planning a European trip, Rough Guides, Lonely Planet, etc. your public library will have a selection if you didn’t want to buy a book. Also, many guidebooks have electronic versions. So, worth checking out.

This will get you started exploring

https://www.ricksteves.com/europe/portugal

Just to add a bit for your perusal. I’ve traveled to Portugal twice now and absolutely love the country. Here are some ideas for some of the places you plan to visit.

Lisbon
Sao Jorge Castle
Explore districts: Alfama, Baixa, Bairro Alto, Chiado
Gulbenkian Museum
Fado Museum
Tile Museum

Belem
Monastery of Jeronimos
National Coach Museum
Monument to the Discoveries
Belem Tower
MAAT (Museum of Art and Architecture)

Porto
Beautiful tile display at train station
Visit wine/porto lodges
Vila Nova de Gaia
Dom Luis 1 Bridge
Stock Exchange Palace
Vintage trolley tour and ride some funiculars and elevators

AutoEurope gets many many recommendations--I would start there. The different drop-off location should be no problem at all.
You do not need an International Drivers Permit for Portugal, but if you think you might venture into Spain, you do need one there (Just FYI).

Thanks for all the advice! It's super helpful, and I appreciate it. I checked out EuroCar, and it looks great. We have to decide when to pick up and drop off the car. From what I've read, having it in Lisbon and Porto might be a hassle. So I think we'll spend a few days in Lisbon and pick up the car when we are ready to head to the Algarve. We'll then drop the car off when we arrive in Porto and take public transportation there. Does this make sense?

That makes sense, and is how I would plan it (have not been to Portugal yet but always researching!). The caveat would be if you are planning to do any day trips in the Porto region.

Thanks for the validation!! I guess I'll check with the hotel we are staying at—Maison Albar—Le Monumental Palace—and find out the parking scoop. If it is not expensive (the hotel is a real treat for us!), maybe it makes sense to keep the car.

Sounds lovely--it'll cost you a pretty penny to park, but it would be worth it if you have day trips in mind that you could not do easily otherwise. I would not have special concern about driving in Porto. You'll be pretty used to driving in old towns by that point.