When I log case activities I often tag in an organisation or individuals name but when I return to the home screen, search for the organisation and check if there are any new activities or relationships listed, there are none. How can I get these two parts of the CRM to work together.
I think by "tag" you mean put their name in the "reported by" field or something like that. Case activities don't appear in "non-case" contexts. This was originally a design goal to both keep case data more private and force people to keep case data inside cases instead of a mix of places.
So the short answer is you need to go to the Manage Case screen to see case activities.