Just take your birth certificate, 2 pictures of yourself that can be done pretty much anywhere...CVS, Walgreens, some Sears Photo places will do it. Check around, prices vary. Bring all that, along with your completed application and the cost of the passport to the Post Office. If you haven't completed the entire application because you are unsure of a certain portion, leave it blank, the person at the PO will help you. Once everything is filled out, you will get a receipt saying it is being sent and the date on it. Then just sit back and wait. It can take up to 6-8 weeks for processing, depending on the demand in the area that it's being sent to at the time. It's a pretty easy procedure. Nothing to really worry about except having the money to pay for it.
Japan is a very safe country. Mom will worry about you, but Moms do that ALL the time! Tell her that there is a lady on the board that lives in Japan...Panda (yumi). She's actually in Tokyo which is north of Nagoya, but I"m sure you could probably catch a train at some point and visit with her. She'd be glad to show you around, I'm sure!
One other thing....since you will be a student there, you will probably need a Student Visa also. A tourist passport will only let you stay in country for a certain amount of months, whereas a Student Visa is good for the school year, then (I think) needs to be renewed every year.