There's a separation between 'inside shoes' and 'outside shoes' in Japan. In particular, there are a variety of places where you will be expected to take off your 'outside shoes' and either switch to 'indoor shoes' or use slippers.
For example, you'll switch to indoor shoes at school and at a gym/sports venue (if you decide to join/use one during your time here).
You should also be prepared to potentially have to take your shoes off and use slippers when going to the bathroom, at traditional Japanese restaurants, bars, and hotels as well as many sightseeing places including inside castles and other historic buildings like shrines and temples.
Keeping that in mind, a general recommendation is to have shoes that are easy to slip in and out of. You may also want to have shoes that cover you in the following situations:
Indoor school shoes (see senpai advice at the bottom of this page)
Indoor sports shoes
Outdoor sports shoes
Outdoor walking shoes that are easy to remove/put on (when you're planning to explore traditional places)
*Hiking boots (If you enjoy hiking, Oita and Kyushu have a lot of options - and if you have a larger shoe size these may be hard to find in Japan.)
Gumboots/Rain boots (Japan has a rainy season. Also, if you're in a rural area and teaching primary school, your students may farm/learn about farming and you may be invited to participate - so again, if you have a larger shoe size, these may be hard to find in Japan.)
See above, but essentially:
wear socks (barefoot in slippers is uncomfortable, particularly when you're hot and sweaty in summer)
be aware that people will see your socks
Over womans' size 8, I would look to bring shoes, some nice slipons/tennis shoes/etc. Really anything is ok as long as it "hasn't been worn outside."
I'm a U.K. Shoe size 8. I fit into the school slippers. However, if you want to be comfy, you should bring your own shoes. If you wait until you are in Japan to buy shoes, expect to have to buy mens' shoes if you are a U.K. Size 6 or above.
If you are over size US 8/8.5, good luck my friend. Some places in Fukuoka have shoes. If you are under, no problem.
If you have big feet, bring your own as Japan sizes for men don't usually go over 28.5cm. Anything over that will be limited.
Anything over a size 10 men's is difficult to find outside of big cities.
Probably bring own shoes if over size 10 US.
For Americans, if you're over size 11, I recommend buying before coming. Indoor shoes should be slip-ons since you'll be coming in and out of the school often.
Shoe Recommendations (for School)
Have ones that are easy to slip on and off.
Cheap and light so you can carry them in your bag (if you have multiple schools).
Crocs/knock off crocks or sandals.
Comfortable slip on loafers or walking shoes that don't stand out, but its almost anything goes.
Slip on sneakers! - Simple, comfy Keds/sneakers that are easy to slip on and off.
Any slip-on shoes are ok, even Crocs (no, really).
Anything that's easy to slip on. A comfortable pair of flats, perhaps. I wear crocs flats that aren't too croc-y.
Slip on sneakers, most of my co-workers wear open toes shoes (with socks).
I use school guest slippers.
Wore sandals. Would have preferred shoes but had already purchased sandals.
Comfortable sandals. I bought mine before coming because I have really wide feet and prefer arch supports.
Black slip on flats or crocs.
I bought Crocs and they have lasted me almost 4 years.
Wait until you get here to buy them, in my opinion. You can buy a 300 yen pair of cheap crocs at COSMOS, then spend a couple weeks at work to see what is okay to wear and what is not. I have kept my cheap black fake crocs the entire time.