First of all, what do you want to study during your time abroad? Japan has a lot to offer after all. Do you want to learn about urban development in the heart of Tokyo or are you more interested in studying Japan's agricultural transformations in Hokkaido? Maybe you want to study Zen Buddhism in the historic temples of Kyoto or live with a Japanese family near Mount Fuji. Any way you want to go, you'll be gaining invaluable experience- there's simply no better way to learn Japanese, or any language for that matter, than by immersing yourself in it. You may find that your single semester in Japan has taught you more than years of study back home! But you already know all this, right? That's why you're reading this now. So think about it- figure out what you want to learn, figure out where you want to learn it and figure out how you want to learn it.
The most popular option for students is to study at a Japanese university, or Daigaku, through an organized study abroad program. There are tons of organizers with offices in the U.S. so communication is easy, support is plentiful, and you're usually pretty certain what you'll be getting. With an organized program much of the legwork is already done for you, which can save tons of time and the headache of organizing your visa or cutting through any other red tape. Be aware though, you will pay a premium for this support. If you're interested in the help though, the end result can be well worth the costs up front.
Depending on your situation, many study programs need not be cost prohibitive. You can obtain federal financial aid for your study abroad program if it is approved by a federal financial aid counselor. If you're interested in learning more about federal financial aid and study abroad you should pay a visit to the financial aid page on the National Association for Foreign Student Advisers site.
You can also win scholarships to help pay for your study abroad experience in Japan; in fact, some federal scholarships and fellowships may be able to cover your costs entirely! There also numerous privately funded scholarships exclusive to American students enrolled full time at a Japanese university. Many of these scholarships will provide you with a travel grant and a stipend for day to day expenses. As is the case whenever free money is involved, competition can be fierce so do your research and get applications off before you need the money, not after. If you can't find much financial aid before you leave you still may be able to obtain a scholarship in Japan, usually you will have to fund your first semester in Japan yourself to do this though. Visit this website for a list of Scholarships you can apply for.