How can we tell when Scripture is talking about one of the persons of the Trinity and when it's talking about all three? I am very interested in learning more about the Trinity. I love God, and really want to understand Them better.

..I mean Him.  This came to my mind earlier, and I laughed, but then caught myself, because Scripture speaks of God in the plural in at least one place:


  Then God said, “Let us make mankind in our image, in our likeness -Genesis 1:26


I realized, this passage must be talking about "The Godhead" which is comprised of the Father, Son and Spirit.  (explained here), 

There are other passages which clearly identify one or all of them distinctly, for example:


  Whoever confesses that Jesus is the Son of God, God abides in him, and he in God. -1 John 4:15


My question is, what about all the other verses (especially in the Old Testament) that are not so obviously talking about the Godhead or one of the Persons?

In light of the doctrine of the Trinity how can we tell if a passage is referring to one of them or all of them?  And if one, how can we tell which one?  For example, is "YHWH" always referring to the Father?
Basically, when the bible talks about God, it means the Trinity or Godhead, while when it says one of their names, such as the Father, it means the Father. But as the Trinity is known--Three in person, One in essence--Then if you talk about the Father, you must be talking in a round-about way, of the Son, and the Holy Spirit, though all are separate.
One link I found, says that there are 4 rules to the Trinity:


There are three persons who have a complete divinity, not a part of divinity, therefore, The Father is a complete God, the Son is a complete God and Holy Spirit is also a Complete God.

Each person is separated from the others, i.e. the Father is not the Son, the Father is not the Holy Spirit, and the Son is neither of them.

There is only one true God,

There is an equality characteristic between the persons of trinity



As the Orthodox Church believes:

Following the Holy Scriptures and the Church Fathers, the Church believes that the Trinity is three divine persons (hypostases) who share one essence (ousia). It is paradoxical to believe thus, but that is how God has revealed himself. All three persons are consubstantial with each other, that is, they are of one essence (homoousios) and coeternal. There never was a time when any of the persons of the Trinity did not exist. God is beyond and before time and yet acts within time, moving and speaking within history.
The source and unity of the Holy Trinity is the Father, from whom the Son is begotten and also from whom the Spirit proceeds. Thus, the Father is both the ground of unity of the Trinity and also of distinction. To try to comprehend unbegottenness (Father), begottenness (Son), or procession (Holy Spirit) leads to insanity, says the holy Gregory the Theologian, and so the Church approaches God in divine mystery, approaching God apophatically, being content to encounter God personally and yet realize the inadequacy of the human mind to comprehend Him.

So don't get exasperated when you can't comprehend the Godhead. God are impossible to fully realize and know. Here is a nice story that shows this (Actually only the first part is the story, but it is all good).
In response to this:

My question is, what about all the other verses (especially in the Old Testament) that are not so obviously talking about The Godhead or one of the Persons?

If it just says God, then take them all, they're all One though Three. If it just says the Father, then take them all, they're all Three though One.
By the way, have you ever read or watched Flatland? Somewhat on topic.