Hi, I’ve just returned from the Med and we visited Valencia (sorry, not the other ports). The shuttles from the ship were free and regular. They drop you off on the outskirts of the City by the Museu de les Ciences. You will be given a map. You need to either get a taxi or a bus from here. It’s very easy and costs €1.50 each on the bus. It takes about 10/15 minutes to reach the centre. You can catch bus number 95 or 35. 35 drops you by the Station and we caught the 95 back as it was a quicker journey. Alternatively you can walk into the city along the old river bed which is now a green open area and looks like a 45/60 minute walk (although we didn’t do this).
Obviously you could get a taxi direct from the port if you prefer.
The daily programme no longer includes much information about the ports, sadly.