From Irun to Hendaye you just take the local "Euskotren" train. That operates every half hour, only costs a couple of Euros, and only takes a few minutes. Its mass transit. Euskotren being a mass transit sytem you do need to change stations. In Irun the Euskotren station is a few blocks away from the RENFE station. In Hendaye the Euskotren station is next to the SNCF station.
These Euskotren trains are currently the only local cross border trains. You just buy tickets at the station.
Given the walk involved in Irun you might want to consider just taking a taxi. it is just a few km.
In Hendaye you can take the SNCF train to Bayonne.
bahn.com actually now has the schedules for that route.
It used to be that Irun/Hendaye was basically treated as a joint station. Spanish trains would terminate in Hendaye, and return empty to Irun. French trains would terminate in Irun, and return empty to Hendaye. That has ended quite a while ago though.