After doing some checking, it seems the best way is to hire a car and driver. Am I missing something,?
Hard to answer since you don't give us any info about what 'best' means for you. (Cost, time, comfort, convenience, etc.) For options, see Rome2Rio.com. Enjoy your trip!
Alas, Spain and Portugal are poorly connected as far a ground transportation is concerned. Often there isn't a good way to connect Spanish City A to Portuguese City B. It is so true that a lot of us discourage travelers from trying to combine the two countries on a single trip unless they have quite a lot of time.
In this case, though, you may be in luck. Check flixbus.com and https://www.alsa.com/en/web/bus/home. Scheduled running times vary. And be sure you use the correct day of the week for your research. On the ALSA website, Porto is known as "Oporto".
We visited Salamanca while on a Douro River cruise. Loved that city.
Renting a car would involve dropping the car in another country and could result in a high charge.
Hiring a car and driver could be expensive as well. I seem to recall that our drive from the Douro River port close to the Spanish border was about 1.5 hours. There is a train that runs up the Douro Valley on the north side of the river. Not sure how often it runs or how far east it goes, but check it out and perhaps you could shorten your ride by picking up the train.
There may be rail service from Salamanca to Porto, check it out. Also, there may be bus service.