“If there be nothing new”
The opening premise echoes Ecclesiastes 1:9 — 'there is no new thing under the sun.' The Teacher of Ecclesiastes argues that all human effort repeats an endless cycle; Shakespeare's poet borrows that mood of world-weary skepticism to question whether his own praise is anything but repetition.
“second burthen of a former child”
'Burthen' is an archaic spelling of 'burden,' meaning a load or weight — but it also means the refrain, the repeated lines of a song. Both meanings are active: the poet is carrying the weight of repeating old praise, and he is singing an old tune over again.
“Since mind at first in character was done”
'Character' here means written letters or inscription, not personality. The line means 'since thought was first set down in writing' — that is, from the very beginning of written history. The poet wishes he could search all of recorded time for an image of the youth.
“Wh'r we are mended, or wh'r better they, Or whether revolution be the same”
'Wh'r' is a contracted form of 'whether,' used twice for rapid rhetorical enumeration. 'Revolution' carries its Elizabethan astronomical sense: the orbital return of a planet to its starting point, and so the turning of time's cycle back to where it began. The three alternatives — we have improved, they were better, or history simply repeats — are the sestet's open question, resolved dismissively by the couplet.
historical The sonnet form