How does one say "to woo" or "to court" (i.e., to solicit someone for marriage) in Latin?
One definition (10.c in Oxford Latin dictionary) of petere is 'to seek the hand of (a woman) in marriage; also ~ere in coniugium or matrimonium.' For example, in Livy, Ab urbe condita 40.4.3, we find:
Theoxena multis petentibus aspernata nuptias est.
Although many were seeking her hand, Theoxena rejected marriage.
The noun for 'suitor' or 'wooer' is procus. In Metamorphoses 13.735, to take one example, Ovid combines the noun procus and the verb petere:
hanc multi petiere proci...
Many suitors courted her...
There's also the related verb procare. that's used in fr. 7 (according to the numbering in Morel's edition) of Livius Andronicus's Odyssia:
matrem <proci> procitum plurimi venerunt.
A great many suitors came to court my mother.