I'm guessing that you have a link to 'page.url_path' within your template. You should use the tag {% pageurl page %} instead:
http://wagtail.readthedocs.org/en/latest/core_components/pages/writing_templates.html#pageurl
Every page, including the homepage, has to have a non-empty 'slug' - I assume this is what you're referring to when you mention setting your home page URL to 'home' - however, if you have a Site record pointing to that homepage (which is how it's set up by default, but see /django-admin/ -> Sites if in doubt) then URLs will be generated based on their paths relative to that homepage, meaning that the homepage itself receives a URL of '/', and a child of it would receive a URL of '/some-child/' despite its location in the tree being '/home/some-child/'. Hope that makes sense!
- Matt