Lines 1-4 Summary

Lines 1-2

Not marble, nor the gilded monuments
Of princes, shall outlive this powerful rhyme;

• Shakes starts out strong, declaring that his poetry is both buffer and more long-lasting than all those fancy funeral monuments of dead rulers. After all, they're dead, right? That's his point exactly.
• "Monuments," meaning the statues and art decorating rich graves, makes it clear that these princes are now past tense, while "marble" and "gilded" reinforces the wealth of these guys.
• "Princes" here doesn't just mean the sons of ruling kings. It's a more general word that refers to every kind of ruler, including queens, dukes, and duchesses.
• And it's set in a really general context. We don't know what country we're in or what century.
• Shakes spreads the irony thickly here by stacking his small flimsy sonnet against the stone art of dead rich guys. These princes thought they could preserve their memories through monuments, but Shakespeare's here to deliver the (not-so) hard truth: time will eventually destroy this stuff.
• As for this sonnet? It will "outlive" everything. How? Precisely because it's not made of hard material. This 14-line iambic package of poetic thought is immaterial: it's not made of anything and therefore can never be destroyed. Boom.
• We've got two pairs of alliteration here, one in each line. "Marble" and "monuments" bring the rich-sounding M's in line 1; the P sounds of "princes" and "powerful" pop out in line 2. 
• Feeling stressed? This baby's got stresses all over it, but only every other syllable. The lines are in iambic pentameter: 10 syllables and 5 stresses per line.
• Head down to "Form and Meter" for more on that.

Lines 3-4

But you shall shine more bright in these contents
Than unswept stone besmear'd with sluttish time.

• You probably came into this assuming it was a love poem, but we don't sight the beloved object until line 3 when "you" sails into the picture with no further introduction. From the other sonnets, however, we know that this "you" is the same as the "fair youth" mentioned elsewhere, a.k.a.: the speaker's hot young man.
• The opening "but" sets these lines in contrast to what we've already read. Marble and monuments are going to topple, but this dude is going to stay bright and beautiful, safe in the house of Shakespeare's sonnets.
• Shakes amps up the brightness by comparing it with another scene of time's destruction: stone floors, probably of a church, that have become filthy and full of trash after many years.
• Church floors were popular places for grave memorials. Wealthy church members were often buried beneath the floor and then memorialized with an inscription on top. The unswept, moldy trash is covering up these plaques, just like time is wrecking the grander grave memorials in lines 1-2.
• "Sluttish" here is probably not sexual—time isn't sleeping around—but just means really disgustingly messy. "Besmear'd" reinforces that idea, bringing a connotation of slime and rot.
• These lines repeat the theme of strong, supposedly permanent things decaying as time goes by. And once again, Sonnet 55 has the last laugh: stone gets trashed but the beloved boy beams brighter than ever
• Get a load of the S and T consonance pileup in line 4 that really makes you hear the mess time's left. On the other hand, the nice long I assonance of "shine" and "bright" in line 3 underlines the eternal beauty of the mysterious "you." For more on those techniques, check out "Sound Check."