doer is the subject.
what the subject does is the verb.

Dogs bark.[Dogs subject. bark verb]
Dog barks.[dog subject barks singular verb]

Every sentence has two components - subject phrase and a verb predicate. A subject defines the topic and a predicate adds a comment about the topic.

So, first of all first try to identify the main verb (signifying tense) and see how other words structure itself to form a clause. What makes a clause is the existence of a finite verb. A subject is the one about whom the verb predicate comments.

The other way is : place the auxiliary Verb ( when there is no auxiliary verb, used do, did..)before the subject, converting it into a interrogative one : if it make sense then it is the subject.

The president will visit Moscow in June.
Will the president Visit Moscow in June ? --- if this makes sense, then the identified subject is correct.

Please note : this can be used only you have a concrete subject , not with some, any.....(SANAM pronoun).(for deciding S-V agreement)
-----

doer is the subject.
what the subject does is the verb.

This is not always true. If I write,

The book is on the table. --- then "the book" is not doing the action. "on the table" is an adverb predicate or some texts refer it as a subject complement.

-----

Hope this helps !!