Old English locian "use the eyes for seeing, gaze, look, behold, spy," from West Germanic *lokjan (source also of Old Saxon lokon "see, look, spy," Middle Dutch loeken "to look," Old High German luogen, German dialectal lugen "to look out"), a word of unknown origin. Breton lagud "eye" has been suggested as a possible cognate.
In Old English, usually with on; the use of at began 14c. As a word to call attention, c. 1200. Meaning "seek, search out" is c. 1300; meaning "to have a certain appearance, express or manifest by looks" is from c. 1400. Of objects, "to face in a certain direction," late 14c. To look like "have the appearance of" is from mid-15c. Look after "take care of" is from late 14c., earlier "to seek" (c. 1300), "to look toward" (c. 1200). Look into "investigate" is from 1580s. To look forward "anticipate" is c. 1600; especially "anticipate with pleasure" from mid-19c. To look over "scrutinize" is from mid-15c.
Look up is from c. 1200 in literal sense "raise the eyes;" as "research in books or papers" from 1690s. To look up to "regard with respect and veneration" is from 1719. To look down upon in the figurative sense "regard as beneath one" is from 1711; to look down one's nose is from 1921. To not look back "make no pauses" is colloquial, first attested 1893. In look sharp (1711) sharp originally was an adverb, "sharply." To look around "search about, look round" is from 1883.