OMIYAGE (GIFTS)
Gift giving is a part of Japanese culture and while a bit confusing at first, you will be able to get the hang of it.  You can give your gifts on the first day, or wait until the start of school in September.  Give omiyage to your supervisors and people who have helped or will help you in the future.  As for how many people you should bring gifts for, if you are a one-shot, the only people you must really cover are your supervisor and the head of your section.  If you are at a base school, you should give a gift to your supervisor, English department head, vice principal(s), and principal.  It is also nice to have a stash of generic gifts in case you give gifts to the wrong person or found that a plethora of people have come to your aid.  Some people also like to get something for all the staff like a box of cookies, find out how many people are in your department, office, or school so you can make sure to bring the right amount.
Don't worry yourself too much, these are small tokens of appreciation not dowries.  Bring something unique that shows off your country or home area/state/province/etc.  When shopping, keep weight and size in mind, as well as the fact that it will be very sticky and hot when you arrive in July. 　Chocolates and candy are a nice gift, but make sure they are packaged to handle the heat.
 

GENERAL OMIYAGE IDEAS
*coffee table book about your country, tea towels, handkerchiefs (they are popular here), scenic calendars, t-shirts, mugs, flavored tea and coffees, jams, soaps, locally made handicrafts, postcards, keychains, bookmarks, lapel pins, deck of cards, fridge magnets, pens, pencils, anything that has a flag, emblem, or the name of  your country/state/town, etc. on it.
*Alcohol goes down well - pick beverages which your country is known for.  You can leave this shopping (unless you plan to bring very local brew) for the airport Duty Free Shop.

AUSTRALIA: wooden animal carvings, two-up sets, boomerangs, Aboriginal related items, Ken Done things, Macadamia chocolates, things with kangaroos or koalas on them.

CANADA: Canadian Rye Whiskey, salmon, maple syrup (but it can be heavy), maple sugar candies, First Nations related works, chocolates/fudge from Purdy`s or Rocky Mountain Chocolate Factory, handmade soap from Lush, things with maple leafs on them.

ENGLAND: Quality Street goods, shortbread, and anything to do with the Royal Family, Shakespeare, British Museum, or the Beatles.

IRELAND: Anything with traditional Irish design-work, pewter-ware, items with a shamrock or four-leaf clover on them.

NEW ZEALAND:  Griffins assorted biscuits, NZ honey, Maori related items, anything to do with kiwis, sheep, or All Blacks.

SCOTLAND: Whiskey, whiskey fudge, small tartan covered boxes of chocolates/fudge, shortbread, needle work, things with Edinburgh Castle, highland dancer, bagpipe player or Nessie on them.

UNITED STATES:  First Nations/American Indian related items, traditional Americana items, most things will be specific to your state ex.) Vermont maple syrup or Ohio buckeye candy, university memorabilia, sports items from NBA, NFL, MLB, etc.