The sonnet begins with a clear statement of its argument: no statue or gold-decorated monument will last as long as this poem. Time will crumble any stone tribute, but poetry can survive forever. And so far, the poem has proved itself right!
The first line is deliberately ornate, subtly mocking the tendency of "princes"—or any person in a position of wealth and power—to try and immortalize themselves. This dense pattern of sound suggests artistry and skillful construction. Hardly a syllable stands alone:
Not marble nor the gilded monuments
Alliteration between "not/nor" and "marble/monuments" (with internal /m/ and /n/ consonance in the latter word, too), consonance on /l/, /d/, and /r/ sounds, and assonant /o/ and /e/ vowels—all in the space of one line! This conspicuous sound patterning mimics the arrogance of these egotistical rulers by being showy and attention-seeking. Like a statue "gilded" with gold, the line itself is flecked with flashy ornamentation.
The enjambment at the end of the first line means the sentence's main verb, "outlive," doesn't arrive until the second line. The delay of the verb first draws the reader's attention to statues and sculptures—the things that it claims poetry (or this poem) will easily outlast. By placing the main elements of the first two lines in this particular order, the poem is doubly dismissive of "marble" and "gilded monuments," mentioning them first so that "powerful rhyme" (poetry) can muscle them out of grammatical place. In short, the poem builds them up and instantly knocks them down.
The second line also uses alliteration and consonance, with the loudest sound being the /p/ of "princes" and "powerful rhyme." That strong /p/ lends drama to the contrast between royalty/power on the one hand and poetry on the other. ("Rhyme" is a synecdoche for poetry: most English poetry in Shakespeare's era was rhymed.) The way the second /p/ falls on "powerful rhyme" seems to steal the power from "princes" and grant it to the poem itself, highlighting where the real power lies.
It's worth noting that line 2 sees the first use of "live" of the poem (in "outlive"). There are three or four of these in the sonnet depending on whether one counts the buried "live" in "oblivious" (line 9). The mention of the word here starts to build a contrast between the living memory of the speaker's beloved (the young man to whom most of Shakespeare's sonnets are addressed) and the dead or soon-to-be-dead artifacts of long-gone "princes" (or other powerful people).