Spain trip in June iternerary

Like you said It looks like trains and buses otherwise you rent a car.
Cuena to Valencia is a Train
Valencia to Barcelona is a bus or train

https://www.alsa.com/en/web/bus/home

You have enough time to sort it out.
You may have to lay out a specific point to point order or rent the car

It appears you’re pretty much visiting the southern and eastern coasts. Not certain how Cuenca fits in since it isn’t near the coast. If you’re doing this off season, then renting a car makes sense since you’ll be able to stop and see things of interest en route. From late May through early September, you could find parking in many of those towns between Cartegena and Barcelona to be a real pain. It’s also a pain in larger cities. I loved Almeria and its Alcazaba fortress. Murcia and Cartegena were nothing special. If you don’t plan on stopping between the places you mentioned, then public transportation may be the way to go.

Start in Cadiz
Train to Sevilla
Train to Malaga
Almunecar as side trip by bus from Malaga
Train via Antequera to Almeria
Bus to Alicante
Train to Cuenca
Train to Valencia
Train via Barcelona to Girona
Train to Barcelona

Thank you David in Brisbane, I will now look up how that works, its exactly what I needed.

I didn’t look fully In to buses from Malaga to Almenucar, and Almeria to Alicante, but i would think they are available for you. Good luck.