How do I display a field 'category' (so as 'tags') in the node.tpl.php?
Every field that is associated to a node is shown in the node view page, if the field is not set to be hidden.
You don't need to change the template file used for nodes, as it shows all the node fields by default. If you are not seeing a field is probably because that field settings are set to not show that field.
As you are referring to the home page (a.k.a. the front page), which normally shows the teaser of nodes promoted to front page, check the field settings for the teaser view mode.