What would I have to do to be eligible to be a teacher in Poland?
You have to do a course which includes teacher training. Every subject studied in schools has such a course. Depending on the course, you either do the teacher training as additional classes, or it's integrated into the course programme - it depends.
Could I be a full time teacher or should I only teach English?
Sure, you can be a full time teacher - all you need to do is complete the programme.
I would want to take the classes in English if that's possible.
Could be more difficult - as far as I'm aware, the only thing which offers teacher training in English will be an English degree.