A chasid quoted his Rebbe; "The ultimate goal of our avodah Hashem (service to G-d) is to attach to G-d." At best this may be a misquote by the chasid, or a lazy choice of words by his Rebbe, but I sadly think that he accurately quoted his teacher. It is often said that performing a mitzvah attaches us to Hashem.
There is no attachment to G-d because there never is detachment from G-d. We must use an accurate description of the goal or we will waste our lives searching for something that does not exist. It is like someone saying over and over again, "I want to be free. I want to be free." He is actually saying, "I am not free. I am not free." All day long he begs for something that he already has. Not only does this fill his head with mistaken information, even worse, his mistake keeps him from ever searching for and attaining the proper goal.
What then of the statement that our sins separate us from G-d? Our sins thicken the veil that hides G-d's revealed Presence from us. Nothing can separate us from the Infinite. The Infinite has no boundaries. If we say that the goal is to attach to G-d we are in fact saying, and embedding into our spiritual heart, that we are detached, i.e. separate from G-d.
G-d is all. There is nothing else.[i] He fills and surrounds the entire creation. He was before time began and is after time ceases. He is our very life and being. The goal is not to attach to what is already All.
Instead, we should be trying to reveal G-d's Presence, or strive to cleave to the revelation of His Presence. We want to bask in the radiance of His Shechina (revealed Presence). We want to know G-d, to serve G-d, to love Him and His ways.
If we remember where G-d is when we pray we do not scream. We speak lovingly to the Holy One Who fills our very being, the One Who is our very breath.
[i] Deuteronomy 4:39