The route that goes through St Jean Pied de Port is the
Camino Frances. If this is your first Camino it is the one I would advise taking. There are many more facilities, much to see and plenty of people to talk to.
Much of the CF is cyclable and as long as you are not going at the height of summer (July August) and are considerate to walkers good cycling. The path sections I would not cycle are the downhills in the mountains. They are steep and narrow and you are likely to have problems as there will be walkers on the same paths. There are quiet road alternatives in every case. Obviously conditions will alter what is cyclable and what is not. A hard pack clay path in summer may be a sticky mess at other times of the year.
Basically follow the yellow arrows and see if the path suits you. If you have a guide-book you will almost certainly be able to see the alternative road section and you can decide then which way to go.
The files on
Santiago Bicicleta give much more information on routes.