I don't understand how to use "where" or "that" when explaining more about the places. - any places where we can walk to - any places that Pudding can run around Does the meaning change if I use it differently? Like "any places that we can walk to" or "any places where Pudding can run around"? Does the speaker just use it by their preference?
The use of "that" and "where" is pretty much interchangeable. As per your example changing the positions of "that" and "where" in this case, would not affect the meaning of the sentence. It is just at the speaker's discretion. Ex: A place that I can play. Ex: A place where I can play.