Old English sceotan "to hurl missiles, cast; strike, hit, push; run, rush; send forth swiftly; wound with missiles" (class II strong verb; past tense sceat, past participle scoten), from Proto-Germanic *skeutanan (source also of Old Saxon skiotan, Old Norse skjota "to shoot with (a weapon); shoot, launch, push, shove quickly," Old Frisian skiata, Middle Dutch skieten, Dutch schieten, Old High German skiozan, German schießen), from PIE root *skeud- "to shoot, chase, throw."
In reference to pool playing, from 1926. Meaning "to strive (for)" is from 1967, American English. Sense of "descend (a river) quickly" is from 1610s. Meaning "to inject by means of a hypodermic needle" is attested from 1914. Meaning "photograph" (especially a movie) is from 1890. As an interjection, an arbitrary euphemistic alteration of shit, it is recorded from 1934.
Shoot the breeze "chat" is attested by 1938 (as shooting the breeze), perhaps originally U.S. military slang. Shoot-'em-up (adj.) in reference to violent entertainment (Western movies, etc.) is from 1942. Shoot to kill is attested from 1867. Shoot the cat "to vomit" is from 1785. To shoot the moon originally meant "depart by night with ones goods to escape back rent" (c. 1823).
O, 'tis cash makes such crowds to the gin shops roam,
And 'tis cash often causes a rumpus at home ;
'Tis when short of cash people oft shoot the moon ;
And 'tis cash always keeps our pipes in tune.
Cash! cash! &c.
["The Melodist and Mirthful Olio, An Elegant Collection of the Most Popular Songs," vol. IV, London, 1829]