care (n.) Look up care at Dictionary.com
Old English caru, cearu "sorrow, anxiety, grief," also "burdens of mind; serious mental attention," from Proto-Germanic *karo (cognates: Old Saxon kara "sorrow;" Old High German chara "wail, lament;" Gothic kara "sorrow, trouble, care;" German Karfreitag "Good Friday"), from PIE root *gar- "cry out, call, scream" (cognates: Irish gairm "shout, cry, call;" see garrulous).

Different sense evolution in related Dutch karig "scanty, frugal," German karg "stingy, scanty." The sense development in English is from "cry" to "lamentation" to "grief." Meaning "charge, oversight, protection" is attested c.1400, the sense in care of in addressing. To take care of "take in hand, do" is from 1580s.
care (v.) Look up care at Dictionary.com
Old English carian, cearian "be anxious, grieve; to feel concern or interest," from Proto-Germanic *karo- "lament," hence "grief, care" (cognates: Old High German charon "to lament," Old Saxon karon "to care, to sorrow"), from the same source as care (n.). OED emphasizes that it is in "no way related to L. cura." Related: Cared; caring.

To not care as a negative dismissal is attested from mid-13c. Phrase couldn't care less is from 1946; could care less in the same sense (with an understood negative) is from 1957. Care also figures in many "similies of indifference" in the form don't care a _____, with the blank filled by fig, pin, button, cent, straw, rush, point, farthing, snap, etc., etc.

Positive senses, such as "have an inclination" (1550s); "have fondness for" (1520s) seem to have developed later as mirrors to the earlier negative ones.