Planning a trip in Europe, my first solo trip. I want one of my stops to be a hike in the Pyrenees, but transportation is stumping me

My current plan is London -> Paris -> hiking in the Pyrenees -> Barcelona -> Surfing in the Algarve region of Portugal -> Lisbon over the course of roughly three weeks. On my way from France to Spain, I want to stop for a day or more to go hiking in the Pyrenees. A quick search made me interested in the trail to Gavarnie falls, supposedly it’s the second highest waterfall in Europe. However, it doesn’t seem to be too accessible with public transportation, and I’m stumped trying to find an interesting trail which I can actually get to with public transportation.

Can you guys help me out with tips or suggestions? Thanks in advance

Edit: thanks for the advice but what I really need is help on transpprtarion to and from the Pyrenees

Planning a trip in Europe, my first solo trip. I want one of my stops to be a hike in the Pyrenees, but transportation is stumping me