Ok, I'm from France and while I've some done hiking in central Pyrenees long time ago, it's not my area.
Bayonne isn't exactly in the Pyrenees, more at the edges.
There's a train line from Bayonne to Saint-Etienne-de-Port area, which is fairly mountainous.
You can check train timetables from the rail company site (also does buses).
www.voyages-sncf.com
(unfortunatly english language version apparently doesn't do secondary network)
About the hike, depends what you want to do, what's your physical condition...
Topographic IGN Maps are pretty good, cost about 10€ and can be found fairly easily in sport shops, bookshops or supermarkets, notably in hiking touristic areas.
Regarding safety, wilderness is far less remote than in the US so it easier to find human presence and/or get rescue, yet these remains steep mountains, so they are not to be taken too lightly, notably be aware of weather.
Can also check this site:
http://www.saintjeanpieddeport-paysbasque-tourisme.com/en.html
They are selling a guide; most likely in French.
You may try to contact them. There also are probably plenty of hiking guides and maps available.
You may have a look at that site:
http://www.pyrenees-rando.fr/PyrOuest.htm
Here is an example of hike in the area:
http://www.pyrenees-rando.fr/Pyrenees/04PyrAtlantiques/01PaysBasque/Iparla/00Iparla.htm
Starts at a train station, end about 10km from Saint-jean-de-port. Can be done on foot, or hitch hiking, maybe there are local buses, or phone a taxi in although that might be quite expensive (like 10-20€

.
Keep in mind I have no specific knowledge of this area. This is only informations I could gather in minutes of internet research. Merely starting points to be explored.
Hope this helps.