How should I go about switching between a third person POV and a first person POV when writing a story?

Italicize the whole section of one of the points of view. It's simple, and you don't need to make a whole new chapter for it.

I personally avoid stories that change between POV's as they seem unbalanced so my answer would ultimately be "don't do it.", however if you were to do this despite my arguments, you would have to change the POV once you enter a separate chapter.

I've read books that simply just have a new POV every chapter no transition

I wouldn't do it. The literal point of view (which is completely different from person) is still the same character's, so you should either continue in close limited third person, or revise what you already wrote when she's silent to first person.

Got to be a separate chapter - it's not unusual and people will pick it up.

One method is to alternate chapters.