Stanza 1 Summary

Get out the microscope, because we’re going through this poem line-by-line.

If I should die, think only this of me,
That there's some corner of a foreign field
That is forever England.

If he dies, the speaker wants people to only think one thing: that there is some "corner" in a foreign country that is "forever England." Hmm. How can a "foreign field" be "forever England"?
If the speaker gets killed in battle and is buried in the field, that spot will be English, in the sense that English bones will be buried there "forever."
Even if the speaker isn't buried in the field, presumably some of his blood would get mixed in with the soil (gross), which also make the field "English," in a way.
The speaker also means that if he dies on the battlefield, that piece of land will be "claimed" by England. Wars are sometimes fought, after all, over land.
Most of Brooke's poetry is about World War I, so it's a safe bet that the "foreign field" here is probably somewhere in continental Europe.
But who is the speaker addressing? His friends? His parents? The reader? His fellow soldiers? Let's read on…
But! Before we do, a quick rhythm alert! It looks like… yup! We've already got some iambic pentameter on our hands here (for more on that, check out "Form and Meter").

[…] There shall be
In that rich earth a richer dust concealed;

The speaker further describes his death. If he dies, a human body—his body—will be buried in the "earth."
"Dust" here refers to the remains of a human body. In many funeral services, the presider will say something to the effect of "ashes to ashes, dust to dust."
In normal speech we might say "there shall be a richer dust concealed in that rich earth," but this is poetry. Poets love to play with word order. (This order also helps to preserve the poem's iambic pentameter. Check out "Form and Meter" for more on that.)
Notice, too, that the word "concealed" is delayed until the very end of the sentence. Concealed refers to burial so perhaps the speaker is avoiding talking about his own, ultimate fate.
Perhaps he is trying to hold our attention until the very end of the line.
"Rich" refers to the quality of the soil. The "richer dust" is the dead soldier, who is more important—"richer"—than just some plot of land.
Another way to look at this is that the dead soldier might also be "richer dust" because he is English, and thus better or "richer" than the land in which he is buried.
While we gather that the speaker is talking about himself, he doesn't explicitly identify himself with the dust. He doesn't say "my richer dust," that is. It's almost like he's trying to avoid thinking too much about his own death and is imagining the death of some generic, unnamed person.

A dust whom England bore, shaped, made aware,
Gave, once, her flowers to love, her ways to roam,

The speaker tells us more about that "dust."
England gave birth do it ("bore"), "shaped" it, made it "aware." England also allowed it to "roam" her "ways" and gave it her "flowers to love." Basically, England here plays the role of the dust's—the soldier's—mother. (Something to think about for later: why there is so much emphasis on the country-as-parent rather than on the "real" mother?)
The second line here can be paraphrased as follows: "England once gave this dust her flowers to love and gave it her ways to roam."
But how do you give dust a gift? Don't forget that dust here still means (checking the title…) The Soldier.
Dust is an interesting word, though. On the one hand, it refers to soil, and points to the soldier's Englishness. He is one with the dust—the land.
On the other hand, the "dust" refers to the dead body, or even the cremated ashes of the dead body. In a way, the speaker is not really talking about a person anymore, just a corpse.

A body of England's, breathing English air,
Washed by the rivers, blest by suns of home.

The speaker reminds us that this dust was a "body of England's" that really experienced all England had to offer: air, rivers, sun—the works!
"Breathing English air" is strange, since it's in present tense. Is the soldier still breathing the air, even though he is dead? Does he die and still imagine he is breathing it? Or, is "breathing English air" just a metaphor for the soldier's Englishness? You know how some people say that the live, eat, and breathe [fill in their favorite thing (like football or model trains) here]? Well, he's so English, he breathes it!