I will be spending 15 days in Portugal and then I want to travel north to the Bordeaux regions of France. I need advice as to what method of transportation is advisable to make that trip?
Rome2Rio.com is a good first step in answering that type of question.
Anything other than flying will be really slow. Renting a car is likely not to be affordable because of the probable extra charge for renting and returning in different countries.
You can use skyscanner.com to look for flights. I don't know when you are traveling. I randomly checked for the month of April and found that there are non-stop flights on Tuesday, Thursday and Saturday. Probably all on Ryanair. As always, check carefully for all the extra charges.
If you have a lot of time and are willing to make an adventure of it, you could travel by train/bus up to Galicia and then across northern Spain. Many interesting cities up that way.