Shoes Off, Tokyo
I would recommend velcro shoes or shoes without laces, which can be easily put on and taken off. You constantly have to take off / put on shoes... a bit of hard work if you have to lace them up every time.
I would also recommend wearing "city" shoes int he city, rather than trainers or hiking boots. In such a fashionable city, we don't need to stand out any more than we do already!
Clothing/Shoes/Weather Gear: The custom of removing one's shoes is still practiced in most Japanese homes. If you plan on visiting a lot of Japanese friends, you may want to consider bringing shoes you can slip on and out of easily. Also, some fancier restaurants may ask you to remove and check your shoes. I regret bringing my Timberland boots.