Can I check this on an airplane?

I went to my grandparent's house in California and turned out to have got a little more clothes than then I can fit, and yes... the clothes are rolled. And I am going shopping some more, I am trying to get school shopping out of the way. I don't think my grandparent's can part with their luggage, so could I bring a big cardboard box for the excess? My flight is on Southwestern..... and when I change planes for the first time ever do I need to get my luggage from under the plane or will they know where to put it? Sorry, I'm a nieve 13 year old. I don't want to have to worry about dragging that around while finding my gate. I am an unaccompanied minor btw. So here are the questions : Can I bring a cardboard box as a checked bag on Southwestern? (First two bags are free), and also do I need to go to one of those swirly things that give out bags to get my luggage when I switch planes or do they send it to the correct plane? THANK YOU GUYS!