Portugal

Rome2Rio is a good first step toward answering this sort of question. The short answer is, "It doesn't look good." Without a car, the quickest trips require multiple modes of transportation.

Thank you for telling me about that app was not aware thank you again.
Pamela

Don't take it as gospel, Pamela, but it does often highlight travel legs that will be difficult to pull off. Every now and then there's bus service Rome2Rio doesn't know about, and its trip duration and fare information often seem a bit imprecise. One thing that's helpful is that if you keep drilling down, you'll usually get to a link to the appropriate bus company and, with luck, the schedule.

Without Rome2Rio, you're sort of stuck with Google, which typically turns up some irrelevant links along with one or two useful ones.

Direct buses run from Lagos to Sevilla in 5.5 hours, 4x/day in summer, and I would hope that Sevilla fits into your itinerary. I don't think that Cadiz or points east have a direct bus service, so everything would go through Sevilla.