Flying With My JRT

I am a student pilot, and I will soon be getting my license. I have seen stories of people bringing their dogs skydiving, and other things of the sort. I am wondering if there are any problems with bringing my dog with me when I go flying places in the future. I have one JRT that I know wouldn't sit through a flight, but I think the other one would. Perhaps there are physical problems with doing this?

Please let me know, and thank you in advance for answering such a strange question.