Old English bettra, earlier betera "of superior quality or excellence," from Proto-Germanic *batizo-, perhaps from PIE *bhad- "good," but Boutkan finds no good IE etymology. For etymology and evolution, see best. Cognate words also have become the comparative adjective of good in the older Germanic languages (Old Frisian betera, Old Saxon betiro, Old Norse betr, Danish bedre, Old High German bezziro, German besser, Gothic batiza). All are comparatives of a positive (Proto-Germanic *bat) which is not in use.
In Middle English the adverbial form commonly was bet, sometimes also was an adjective; bet was displaced by c. 1600. From late Old English as "improved in health, more healthy" (adv.); from late 12c. as "more useful or desirable." Better half "wife" is first attested 1570s.
c. 1200, from Old Norse geta (past tense gatum, past participle getenn) "to obtain, reach; to be able to; to beget; to learn; to be pleased with," a word of very broad meaning, often used almost as an auxilliary verb, also frequently in phrases (such as geta rett "to guess right"). This is from Proto-Germanic *getan (source also of Old Swedish gissa "to guess," literally "to try to get"), from PIE root *ghend- "to seize, take."
Old English, as well as Dutch and Frisian, had the verb almost exclusively in compounds (such as begietan, "to beget;" forgietan "to forget"). Vestiges of an Old English cognate *gietan remain obliquely in modern past participle gotten and original past tense gat, also Biblical begat.
In compound phrases with have and had it is grammatically redundant, but often usefully indicates possession, obligation, or necessity, or gives emphasis. The word and phrases built on it take up 29 columns in the OED 2nd edition; Century Dictionary lists seven distinct senses for to get up.
"I GOT on Horseback within ten Minutes after I received your Letter. When I GOT to Canterbury I GOT a Chaise for Town. But I GOT wet through before I GOT to Canterbury, and I HAVE GOT such a Cold as I shall not be able to GET rid of in a Hurry. I GOT to the Treasury about Noon, but first of all I GOT shaved and drest. I soon GOT into the Secret of GETTING a Memorial before the Board, but I could not GET an Answer then, however I GOT Intelligence from the Messenger that I should most likely GET one the next Morning. As soon as I GOT back to my Inn, I GOT my Supper, and GOT to Bed, it was not long before I GOT to Sleep. When I GOT up in the Morning, I GOT my Breakfast, and then GOT myself drest, that I might GET out in Time to GET an Answer to my Memorial. As soon as I GOT it, I GOT into the Chaise, and GOT to Canterbury by three: and about Tea Time, I GOT Home. I HAVE GOT No thing particular for you, and so Adieu." [Philip Withers, "Aristarchus, or the Principles of Composition," London, 1789, illustrating the widespread use of the verb in Modern English]
As a command to "go, be off" by 1864, American English. Meaning "to seize mentally, grasp" is from 1892. Get wind of "become acquainted with" is from 1840, from earlier to get wind "to get out, become known" (1722). To get drunk is from 1660s; to get religion is from 1772; to get better "recover health" is from 1776. To get ready "prepare oneself" is from 1890; to get going "begin, start doing something" is by 1869 in American English; get busy "go into action, begin operation" is from 1904. Get lost as a command to go away is by 1947. To get ahead "make progress" is from 1807. To get to (someone) "vex, fret, obsess" is by 1961, American English (get alone as "to puzzle, trouble, annoy" is by 1867, American English). To get out of hand originally (1765) meant "to advance beyond the need for guidance;" sense of "to break free, run wild" is from 1892, from horsemanship. To get on (someone's) nerves is attested by 1970.