A car is best for getting around the Basque Coast and the Pyrénées. Train and bus connections will get you from Bayonne to Biarritz and Hendaye and up to St-Jean-Pied-de-Port easily, but less pivotal destinations will entail much waiting and loss of valuable time. The roads are good, albeit slow, and you shouldn't plan on doing much better than 60 km-70 km an hour on average.
For walking the hills or long-distance hikes across the GR10 or the Haute Randonnée Pyrénéenne along the crest of the cordillera, bus and trains such as the SNCF Bayonne to St-Jean-Pied-de-Port connection will drop you off and pick you up at trailheads such as the one at Bidarrai's Pont d'Enfer.
Bus lines from St-Jean-Pied-de-Port will take you east to Larrau, Mauleon, and Pau. From Pau there are SNCF connections up into the Pyrénées, with subsequent SNCF buses to points such as Gavarnie.