In reply to I like climbing:
> Make sure you get proper instruction - trying to follow a video won't work.
> Good luck.
This is very very true! It will be very hard to learn the correct breathing techniques and good exercise form without input from someone experienced.
Also, there is no central governing or certification body for Pilates instructors so there is also a very wide variety in what is called Pilates today. Some schools and instructors have diverged quite a lot from the original work of Joseph Pilates.
While most/all versions will work your core, the actual execution may range from almost yoga like static positions to pretty hard-core calisthenic and body weight moves.
In my personal opinion find a class/instructor that emphasizes basic mat work. Avoid classes with lots of gimmicks like rings, rollers, balls and bands. These exercises may look simple but they are actually fairly advanced and difficult to execute correctly.
I have been doing Pilates now for 5 years and it has done wonders for my back pain and improved my core and flexibility. As a result, my climbing has improved as well.
Full disclosure, I have Peak Pilates basic mat instructor certification but I am still very very far from event attempting some advanced traditional exercises, let alone teaching it.
Hope this helps.