My main question is this: What is the LDS view of the Trinity? As a Catholic, I believe that the Trinity is One Being (Father, Son, Holy Spirit) in three persons (consubstantial). I guess my question boils down to this, do they (LDS) believe that the Trinity is One, or three separate beings?
The short answer:
Three separate and distinct beings.
In the Doctrine and Covenants (LDS cannon), it states:
22 The Father has a body of flesh and bones as tangible as man’s; the Son also; but the Holy Ghost has not a body of flesh and bones, but is a personage of Spirit. Were it not so, the Holy Ghost could not dwell in us.
Additionally, in Joseph Smith-History he recounts
17 It no sooner appeared than I found myself delivered from the enemy which held me bound. When the light rested upon me I saw two Personages, whose brightness and glory defy all description, standing above me in the air. One of them spake unto me, calling me by name and said, pointing to the other—This is My Beloved Son. Hear Him!
David's link also has good information.
Put simply, they don't believe in it, not the way mainstream Christianity does. They believe in all three entities, but that all three entities are distinct, separate beings.
They refer to the three as "The Godhead". There's more here: http://www.lds.org/topics/godhead?lang=eng
The true doctrine of the Godhead was lost in the apostasy that followed the Savior's mortal ministry and the deaths of His Apostles. This doctrine began to be restored when 14-year-old Joseph Smith received his First Vision (see Joseph Smith—History 1:17). From the Prophet's account of the First Vision and from his other teachings, we know that the members of the Godhead are three separate beings. The Father and the Son have tangible bodies of flesh and bones, and the Holy Ghost is a personage of spirit (see D&C 130:22).