Mormons do not become there own God, our God is always and will always be Heavenly Father. We believe that worthy couples sealed for time an eternity in the temple can be exalted to Godhood, they become Gods through the Atonement.  
 
The process of exaltation and many other details about the afterlife though are very vague, we know that we can become like Heavenly Father but that is about all the scriptures clearly discuss. There is various theories and beliefs surrounding exaltation held by various members but the scriptures still remain unclear.  
 
The difference between the Godhead and the Trinity is that one (Godhead) comes from the Bible and the other (Trinity) has no scriptural backing and is purely the concept of man that uses a key verses taken out of context to prop it's poor claims us.  
 
The Godhead of the Bible states that the Heavenly Father, Jesus Christ, and the Holy Ghost are three individual beings united in purpose as the One True God. The scriptures consistently refer to Christ, Heavenly Father, and the Holy Ghost as independent beings. They are one God though because they are perfect and being perfect they are in perfect union with each other. This union is called the Godhead, Heavenly Father reigns supreme over the Godhead.  
 
In many other mainstream Christian faiths they believe in the Trinity, that Father, Son, and Holy Ghost are 1 being with three personalities, more modernly seeing that this first concept is silly they have come to evolve the Trinity into a 1 beings with 3 manifestations, 3 beings that combine to make 1 God, and 3 beings that cooperate to be one God.  
 
Either way you look at it it is hogwash, IMO. The scriptures clearly define the nature of the Godhead and it isn't the Trinity, IMO.  
 
You can find more information at lds.org and mormon.org  
 
Hope that my answer helped!

Doctrine and Covenants 130:22 states... 
"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." 
 
This is the most plain statement we have on the nature of the Godhead. It is simple. We believe that God the Father, Jesus Christ and the Holy Ghost are three distinct beings beings, yet, being separate, they are One in power, glory, dominion, etc. They are perfectly united. When Joseph Smith received his first vision, he saw two personages before him: God the Father and Jesus Christ. When Christ was baptised, the Holy Spirit descended in the form of a dove and the voice of the Father exclaimed "This is my beloved son." In John 17:22, Christ prays to the Father regarding his disciples "And the glory which thou gavest me I have given them; that they may be one, even as we are one." These are just some of the many evidences that the Godhead is comprised of 3 separate but unified beings. 
 
I can't really explain much about the Trinity because I don't really understand that doctrine. It makes no sense to me, frankly. 
 
Now, God the Father is the literal father of our spirits. We lived with Him before we came to this earth as his spirit children. We were sent to earth to gain a physical body (like our Heavenly Father) and to gain experience and learn to choose right from wrong (also like our Father). The Father knew that on this earth, we would all fall short and make mistakes, thus making us impure and unable to return to His presence. Because of this, He provided us a Savior, Jesus Christ. Christ was also a spirit child of the Father, his first born. He was distinct in that somehow he had obtained the status of God even before coming to this earth. He was like the Father in every way except that he did not yet have a physical body. 
Christ was sent to earth as both the firstborn spirit son of God and the only begotten in the flesh, meaning God the Father was the father of his spirit and his physical body. As such, he had the ability to lay down his life, but also the ability to take it again in resurection. He also had the ability to endure the pains and sufferings of all mankind without dying, which he did in the Garden of Gethsemane. When he suffered for us, he purchased us with His blood and became our Father, making himself both the Father and the Son. Through His sacrifice, all those who believe on him and follow His teachings may now overcome sin and death and return to live with our Father in Heaven, inheriting all that the Father has, being joint heirs with Christ. 
 
Although many people think it crazy or even heresy the idea that one can become like our Heavenly Father, it is actually not unique to our faith. It is my understanding that this doctrine was had among many of the early Christians as well. What you need to understand is that it not that man is rising to be like God, it is God raising man to be like Him. It is His work, not ours. It is what makes Him so great, not us. 
There is a scripture in the LDS cannon that states: 
"For behold, this is my work and my glory - to bring to pass the immortality and eternal life of man." (Moses 1:39)

The Trinity as defined by the Nicene creed defines God as a single being with three different personages. It is considered one of the sacred mysteries of the Christian church.  
 
In the LDS faith, it is not quite as mysterious.  
 
There is God the Father. He's our all powerful Father in Heaven. He has full rights, privileges and power over all things related to us.  
 
There is Jesus Christ, the Son of God. If I can use the term, He has been given stewardship of all things related to us. So while God is God and nothing will change that, Christ is still the man in charge.  
 
And then there is the Holy Ghost which knits all things together into a whole. It is through him that we receive communication and assistance. 
 
These three act as one in all things, but are three distinct individuals. 
 
As far as becoming Gods, we believe that we can learn and progress to become like our Father in Heaven. It is not an instantaneous thing, but something that happens one piece at a time after the work of the earth is completed.