There are no buses from Paris to Bayonne, you will need to take the train. And I really suggest you book a ticket in advance. Is there any chance you can stay in Paris for a couple of nights, you'll be really jetlagged and it's not really a good idea to begin walking straightaway. Or at least one night, just in case your plane is delayed, which could result in you missing the train.
The train to Bayonne leaves from Gare Montparnasse. If you're going straight from the airport to the station, the easiest way is to take the Air France shuttle bus. This costs about 17euros, and goes directly to Gare de Lyon and then Gare Montparnasse. The trip takes around 50 mins. You'll find the bus outside whichever CDG Terminal you arrive at or, if you're not sure where to find it, ask at the customer service desk - they speak English.
When booking your train make sure to allow the time it will take to clear Immigration at CDG, pick up your baggage, find the bus, and get to Gare Montparnasse. CDG passport control is notoriously slow, so allow three to four hours between plane arrival and train departure, more if you can.
At Bayonne station you then get the local train to SJPP.