Do American CDs work with Japanese CD players?
CD's work, but DVD's don't unless you have an international DVD player. Japanese people give each other food (snacks, fruit...pretty good or special snacks/food). since you seem to be coming from another country, it's also normal to just bring something special from where you come or perhaps a simple present. if the people have children, buy some small toys. and maybe get the host parents chocolate (share with everyone?).
Do Japanese CDs work on US players?
Unless the disk is advanced for blue-ray (as in it also contains a video), your CD player is more than 7 years old or the CD is specified beyond just 'music', it should work. Japan uses the same CD burning system as the USA. If it works you can ignore the rest of this info, if not... If your CD doesn't work, put it in your computer, you may have to format the CD for older CD players. (remove videos or/and data files that don't end in an MP3 file name. ( song5.mp3 pizza.mp3 ) If the files all end in MP4, use a converting program to make it an mp3. I've actually had to do the above for a Gackt CD from Japan. It also had a movie clip at the end, so that made the CD too advanced for my old CD player.
Play along songs with different patterns, pay a lot of attention to the pattern, let it become embedded into your muscle and music memory. Try some different songs and patterns every day.