Getting an NPC to follow you around the map means adding them to your party. NPCs with a disposition of 80 or higher (loyal) can be recruited. You can have up to 3 party members at a time.
When you hover over a loyal NPC's avatar in a building, a purple Add to Party button appears above them. Click it to recruit. From then on they follow you wherever you go until you dismiss them.
If the button isn't there, the NPC isn't loyal enough yet — see Disposition for how to raise it.
To dismiss a party member, hover over their avatar on the map or in chat and click Remove from Party. They return to their home location.
Combat alongside party members and auto-join/leave are planned, not live yet.