I am preparing a change set to deploy an updated trigger from sandbox to production. Now, there are associated classes and other dependencies. Do I have to add all the dependencies in the 'Add dependencies' section along with it ? This is my first time using a change set.
No, a changeset isn't any different from any other deployment method in this regard.
If you have code that uses a particular custom field, that field will (or should) show up as a dependency. If that field is already in your target org, then there's no need to deploy it again.
It boils down to "if there is a change between pieces of metadata in 2 orgs, deploy the things that changed".