What exactly is going on in the poem?
One key to untangling this poem lies in understanding the purpose of the four kinds of men the speaker mentions. Each has some epiphany of wisdom or of recognition of failure as they approach their dying end.
The "wise men" realize their words "forked no lightning," or made no lasting earth-shaking impression.
The "Good men" realized that their deeds could been oh so great if they had been somewhere other than where they were: "in a green bay."
The "Wild men" realized their song of celebration of life ("sun in flight") was but a song of grief as all must depart from life.
The "Grave men," whom I take to represent scientists and such ("meteors"), realize they might have been both "grave" and happy.
The speaker is pleading with his father--who seems to have given up beneath the burden of his own revelations of failure and loss--to act like these four kinds of men, similarly burdened with failure and loss, and "rage, rage"--despite his failures and losses in life--and fight for the prolongation of the last light of life before the "dying of the light" takes him "into that good night" of death.
“Do Not Go Gentle Into That Good Night” is an exhortation by the poet for his dying father to embrace every single breath rather than accepting death quietly. The writer does not want his father to submit to death, but rather to “rage, rage against the dying of the light,” urging him to fight to the very last. The poet believes that the old should cling to life as ferociously as any young person would do. He describes the philosophical approaches of four different old men as they approach death.
I think the most important message of this poem is the way that we are supposed to meet and face death. Dylan's elegy to his father stridently demands that we do not meet death meekly and without a fight. It presents a picture of death that suggests that we need to fight for every single moment of our existence and every breath we draw. Yes, death is inevitable, but we should not make its victory over us too easy, the poem seems to argue.
I think that you will find examining the enotes group on the poem will be quite beneficial. In terms of looking at meaning, examine the line "Rage, rage against the dying of the light." It is the most repeated line in the poem and expresses Thomas' feelings about loss, the predicament of death, as well as the activism and futility of human freedom. This might be a good place to begin when analyzing the poem.
Dylan Thomas exhorts his father to "rage against the dying of the light" because it will demonstrate to his son that he does not wish to leave the living. This is a demonstration that Thomas indicates in his poem that he so desperately wants from his father. It is ironic, indeed, that Thomas himself died without "raging" and so young--age 39 when he was in the United States in 1953.
I think that the poem can be extended as a conversation about anyone who is dying. You should not just die. You should fight death until the end, even though it may not seem wise to do so. Life is too precious. The speaker is angry and wants to fight death.