Is it a good idea to go for multiple bachelor's or master's degree? I have a bachelor's degree in Physics. I am currently doing my master's in Physics with a focus on machine learning. I want two thing for future:

I want to gain coursework experience from different country - I am fascinated how coursework, be it undergraduate or graduate level, is done in different countries and I want to experience that.
I want to be a veterinary professional. (I wanted to teach at one point and do research. Therefore, had a plan for PhD but I no longer want to be in academia). That could be a veterinary nurse or a doctor.

I didn't share this with anyone as I was afraid of being judged as immature or naive. I would really appreciate any advice on what I can do to achieve my goals. Are my goals justified?
You need to identify what specific courses are needed to enter into the fields you are interested in. Then determine what you are missing and what your options are for completion.  For example, for many health professions in the US there are post-baccalaureate programs specifically for career changers to finish their prerequisites.