Spain to Portugal by Train

There is a very limited cross-border service between Spain and Portugal.
There is an overnight train Madrid to Lisbon, details here: https://www.seat61.com/international-trains/trains-from-Madrid.htm#Spain-Portugal
There is also a route via Vigo to Oporto, in the very north of Portugal.
For other info on Portuguese trains see here: https://www.seat61.com/Portugal.htm

Ground transportation links between Spain and Portugal are surprisingly limited and run extremely infrequently.

There's one train per day between Madrid and Lisbon. It takes over 11 hours. Chris has pointed out the night train on that route; for some reason it doesn't show up on the Deutsche Bahn webpage that I use for most of my train-schedule research. You can travel from Porto to Santiago de Compostela by changing trains (and train stations) in Vigo. This is also infrequent.

There's bus service from Seville to the Algarve (perhaps Faro?) and from Madrid to Lisbon. There's also some service between Porto and Santiago de Compostela.

It's simpler to fly. Check skyscanner.com for flight options between the two countries.

You can use Rome2Rio.com to see your options at a high level. Do not trust the fares, travel times or frequencies it displays, because they can be wildly off base. Just keep clicking through until you find a link to the appropriate bus or train company's website. Go there for specifics.

Where to go in Portugal depends on what you're interested in seeing and doing, the time of year you'll be traveling, how much time you have, and what is practical from your starting point in Spain.

Vueling Airlines is the budget air carrier that services Spain and Portugal well. I would suggest you check their fares online.
Travel on the ground is often time consuming as Spain and Portugal together are larger than most people realize. When I travel to new places, I look at Google Maps to get the lay of the land--and distances between cities.