What is the cheapest (yet still reasonably comfortable) method to
travel between these Spanish cities?
As you asked for the cheapest way, in my experience that would be the bus; those are generally cheaper than trains in Spain. (What surprised me is that on some routes the bus is also faster than train.) Connections are frequent and the buses are generally very comfortable.
Granted, it was back in 2005 when I travelled more in Spain (including Andalusia), but I suppose the situation hasn't changed too much.
The train answer mentioned Madrid to Málaga as an example (cheapest fare 35 € when booked in advance). On ALSA, the trip costs 23-24 € (no advance booking required). As another example, 8 hrs night bus from Cádiz to Madrid is just 25 € with Socibus.
Photo credit: http://autocaresdemadrid.blogspot.com/
There are several bus companies (partial list) operating in southern Spain. I don't know if there's one website to see all timetables though. What I did was simply go to the bus station in each city or town, check timetables there & buy the ticket; this worked fine. (Or google e.g. "sevilla cadiz bus" to find relevant company website with timetables.)
Btw, while you're in Andalusia, I'd recommend Cádiz, on the Costa de la Luz, less than 2-hour (12.20 €) bus trip from Seville (though not exactly on your planned route).