@kmccue You don't say where you are coming from. That would be helpful to respond to your question.
Many international flights to Lyon may require a change of planes. If this is your case, your overall travel time to Lyon and continuing onward by train to Le Puy might be just as long as flying to Paris and starting train travel there.
French trains are fast and comfortable, though disruptions occasionally occur as biarritzdon just mentioned. I have personally bought my tickets online from Trainline ever since SNCF (the French national railway company) stopped accepting my U.S. credit card some years ago. The Trainline site is very easy to use and they have responded promptly when I have asked them my questions. I think their ticket prices are the same as SNCF's, or at least they used to be.
If you opt for Lyon, the Rhonexpress travels multiple times per hour between the airport and the main Lyon train station where you would continue your journey to Le Puy.
Also, I seem to recall that the most prolific and most grounded poster on this forum (falcon269) said that it is also possible to take a bus from the Lyon airport to Le Puy. There may be a transfer required - seems likely, but I'm not sure.
From Paris by train - you'll be on three trains if you can catch the first from the airport station (CDG). Otherwise you'll have to travel into Paris to Paris-Bercy or to the Gare de Lyon. Possible using public transport but doing so can entail a bit of a hassle.
From Lyon - Rhonexpress plus two trains
Bon chemin.