1530s, "a running (usually at full speed), a course" (especially of the sun, etc., across the sky), from French carriere "road, racecourse" (16c.), from Old Provençal or Italian carriera, from Vulgar Latin *(via) cararia "carriage (road), track for wheeled vehicles," from Latin carrus "chariot" (see car). Sense of "general course of action or movement" is from 1590s, hence "course of one's public or professional life" (1803).
1590s, "to turn a ship on its side" (with the keel exposed, for inspection, repairs, etc.), from French cariner, literally "to expose a ship's keel," from French carene "keel" (16c.), from Italian (Genoese dialect) carena, from Latin carina "keel of a ship," also (and perhaps originally) "nutshell," possibly from PIE root *kar- "hard."
Intransitive sense of "to lean, to tilt" is from 1763 of ships; in general use by 1883. In sense "to rush headlong," confused with career (v.) at least since 1923. [To career is to move rapidly; to careen is to lurch from side to side, often while moving rapidly.] Earlier figurative uses of careen were "to be laid up; to rest." Related: Careened; careening.
"a course, especially a fixed course of study at a college, university, or school," 1824, from a Modern Latin transferred use of classical Latin curriculum "a running, course, career" (also "a fast chariot, racing car"), from currere "to run" (from PIE root *kers- "to run"). Used in English as a Latin word since 1630s at Scottish universities.