I've decided not to start the commentary with an attempt to give a meaning of the opening lines. While my later commentary will address the topic of the meaning of the poem (and other topics that strike my fancy,) I think it is important to emphasis the fact that The Waste Land is a poem, not an essay.
That said, what is of poetic value here? In what ways does Eliot use words to amuse, entertain, and stimulate us?
First is the surprise value of strange connections. April is the cruellest month? Winter kept us warm? While the opening line has become so well known the novelity and suprise has been lost, please imagine buying the issue of Dial magazine as it came out and reading this poem for the first time.
Secondly, with the use of a great story telling mechanism, we are also brought into the poem early on with the line "Winter kept us warm." In the draft version of the poem the opening line was "First we had a couple of feelers down at Tom's place." With this line the "we" does not include you and I. We were not at Tom's place. With the deletion of the original opening lines, the next use of the first personal plural is the ambigous "Winter kept us warm." Is Eliot including us, the readers, as part of the group? Or does he have a different set or group of people in mind? We are not sure at this point. As readers we can stop and think about it but as listeners of the spoken poem we must continue along with the narrator or drop out. Since we have found that we may be part of the poem we agree to join the narrator with his story.
Another thing to notice in the opening lines is the lack of rhyme. Eliot uses rhyme later in the poem. It adds emphasis to selected sections and changes the tone of the poem. Here, instead of our listening for rhyme to follow the flow of the narrator's lines, we listen for his use of the -ing verbs.
And that is another surprise Eliot has given us, he has put emphasis on the words breeding, mixing, stirring and feeding by placing them on the end of the line and not at the beginning as the lines might be written by another poet.
He also has put emphasis on actions (verbs), not on things (nouns); His emphasis is not on month, land, desire, snow and tubers but on breeding, mixing, stirring and feeding.
Also note Eliot's ommission of a noun with his phrases. What "Breed[s] lilacs out of the dead land?" April of course, but we do not see (or hear) the lines "April is the cruellest month, / it breeds lilacs out of the dead land," etc.
Examine some of Eliot's images: "dead land," "dull roots," "forgetful snow." Note some of the strange phrasing he uses: "Breeding lilacs" "Covering earth" (not "the earth,)" "Feeding a little life."
Note also that because of Eliot's phrasing the word "earth" is placed at the beginning of line 6 and is thus capitalized. We are not quite sure whether Eliot means the planet Earth or earth, as in soil. Thus, by line 6 we have already had two ambiguities in the poem, "us" and "earth". There will be many more ambiguities as we continue. They make reading the poem challenging (and frustrating) but give the poem much of its power.
If you wish to continue with this topic, on a questions for the reader page I ask for a comparison the sound of the poem as written by Eliot with an alternate version written in the standard manner.