Public transport links between Spain and Portugal are almost exclusively restricted at present to long distance coaches.
While you will not need a car in Seville, you will need one to explore the Algarve and you may well miss places you want to see in Spain.
I would therefore strongly recommend flying in and out of Lisbon on this trip. Rent a car in Lisbon and drop it off there. Intra-country drop offs are very expensive in Europe. Be aware to specify an automatic if you are from the other side of the pond as in Europe we mainly drive manual shift vehicles. I also suggest a vehicle with aircon as Andalucia will be very very hot at that time of year. (Between 35º and 40º during daylight hours.)
You routing should be by toll road from Lisbon Airport via the Vasco da Gama Bridge to Albufeira. (+- 2:30 mins). Albufeira - Seville freeway (2 hours). On the Seville to Malaga route there are various options, including the scenic long one via Jerez/Algecira/Marbella.
While in Malaga, with a car you can also make an easy day trip to Granada and the Sierra Nevada, although it will be very hot in the former.
To do all the above by bus would at least double travel time and add a lot of inconvenience.
