Old English leornian "to get knowledge, be cultivated; study, read, think about," from Proto-Germanic *lisnojanan (cognates: Old Frisian lernia, Middle Dutch leeren, Dutch leren, Old High German lernen, German lernen "to learn," Gothic lais "I know"), with a base sense of "to follow or find the track," from PIE root *lois- "furrow, track." It is related to German Gleis "track," and to Old English læst "sole of the foot" (see last (n.1)).
From c. 1200 as "to hear of, ascertain." Transitive use (He learned me (how) to read), now considered vulgar (except in reflexive expressions, I learn English), was acceptable from c. 1200 until early 19c. It is preserved in past-participle adjective learned "having knowledge gained by study." Old English also had læran "to teach" (see lere). Related: Learning.
"having knowledge gained by study," mid-14c., past-participle adjective from learn (v.) in its former transitive sense. Related: Learnedly; learnedness. "[L]earned expresses depth and fullness in the knowledge, while scholarly expresses accuracy" [Century Dictionary].
Old English leornung "study, action of acquiring knowledge," verbal noun from leornian (see learn). Meaning "knowledge acquired by systematic study, extensive literary and scientific culture" is from mid-14c. Learning curve attested by 1907.
Old English lar "learning, what is taught, knowledge, science, doctrine; art or act of teaching," from Proto-Germanic *laisti- (compare Old Saxon lera, Old Frisian lare, Middle Dutch lere, Dutch leer, Old High German lera, German Lehre "teaching, precept, doctrine"), from PIE root *lois- "furrow, track;" compare learn.
Old English tæcan (past tense tæhte, past participle tæht) "to show, point out, declare, demonstrate," also "to give instruction, train, assign, direct; warn; persuade," from Proto-Germanic *taikijan "to show" (source also of Old High German zihan, German zeihen "to accuse," Gothic ga-teihan "to announce"), from PIE root *deik- "to show, point out." Related to Old English tacen, tacn "sign, mark" (see token). Related: Taught; teaching.
Lemonade Vendor (Edgar Kennedy), enraged: I'll teach you to kick me!
Chico: you don't have to teach me, I know how. [kicks him]
The usual sense of Old English tæcan was "show, declare, warn, persuade" (compare German zeigen "to show," from the same root); while the Old English word for "to teach, instruct, guide" was more commonly læran, source of modern learn and lore.