Madrid to Lisbon

Sorry to say it, but: Fly. Check skyscanner.com for schedules and approximate fares, but go to the websites of the operating airlines for precise information, including luggage limits and other gotchas.

Driving is out unless you're making a loop trip, because picking up a car in Spain and dropping it off in Portugal usually comes with a very high (at least hundreds of euros) international drop charge.

There are no direct rail links between the two capital cities, or indeed across that border elswhere. It's hard to believe, but true. This is why we usually discourage people for trying to combine Spain and Portugal on the same trip unless they have a lot of time.

You'll find nothing on the Spanish rail site (Renfe), which is a clue that this will not go well. The Trainline, a third-party seller, has nothing a sane person would consider. The Deutsche Bahn (German rail website) actually produces information showing it's possible to cross the border via a train-bus-train combination. You'd be buying the components separately, it seems, and things could certainly go wrong. There are several slightly different routes shown on the DB website, the fastest of which (on a weekday) works like this:

08:51 Depart Madrid-Atocha Cercanias by train IC 190 (seat reservation required)
13:39 Arrive Badajoz

14:09 Depart Badajoz via R482, which seems to be a bus operated by the Portuguese train company
15:51 Arrive Entroncamento

15:59 Depart Entroncamento by train IC 512 (seat reservation required)
16:52 Arrive Lisboa-Santa Apollonia (or maybe Lisboa-Oriente; it's fuzzy)

Those connection times don't inspire confidence in me. Let's just say it would be an adventure.

Here's the Deutsche Bahn website: https://int.bahn.de/en/trains

It's easier to cross the border in the south via bus from Seville to Faro or in the north via (I believe) multiple trains along the Santiago de Compostela-to-Porto route, but it's still nothing like crossing most other borders in western Europe.