At the beginning of the year, we started studying the Teachings of Presidents of the Church: Joseph Fielding Smith manual in our priesthood and Relief Society meetings. I'm trying to catch up with the study schedule, so I recently read the first chapter. I am so impressed with his teachings. There is a reason he was called the "defender of the faith."
Here is a quote from that first chapter that I thought explained our doctrine of the Godhead being three separate personages very well:
"The [first] vision of Joseph Smith made it clear that the Father and the Son are separate personages, having bodies as tangible as the body of man. It was further revealed to him that the Holy Ghost is a personage of Spirit, distinct and separate from the personalities of the Father and the Son [D&C 130:22]. This all-important truth staggered the world; yet, when we consider the clear expressions of holy writ, it is a most astounding and wonderful fact that man could have gone so far astray. The Savior said, 'My Father is greater than I;' [John 14:28] and he invited his disciples, after his resurrection, to handle him and see that it was he, for, said he, 'A spirit hath not flesh and bones, as ye see me have.' [Luke 24:39.] The apostles clearly understood the distinct entities of the Father, Son, and Holy Ghost, to which they constantly refer in their epistles; and Paul informed the Corinthians of the fact that when all tings are subjected to the Father, 'then shall the Son also himself be subject unto him that put all things under him, that God may be all in all.' [1 Corinthians 15:28]" (pg. 37)
Just out of curiosity, I decided to do some research for scriptures and ancient text that that showed the Father, Son and Holy Ghost being distinct beings. I found a few:
John 17:21 (might help clear up any confusion about the term "being one")
Luke 23:34 (this isn't the only time that Jesus prays to his Father, giving us the idea the Father must be a separate being. If they were the same, he wouldn't have had to pray to Him, would he?)
I also know that in the Hebrew text of the King James version of the Bible, it is clear that there is Elohim and Jehovah, and they aren't the same being. Even in the Nicene Creed of 325 A.D., they admit that Christ was begotten of the Father and that He is seated on His right hand. He'd have to be a distinct being to do that. Even the Holy Ghost sounds as though it was defined as a distinct being.