fast (adj.) Look up fast at Dictionary.com
Old English fæst "firmly fixed, steadfast, secure, enclosed," probably from Proto-Germanic *fastuz (cognates: Old Frisian fest, Old Norse fastr, Dutch vast, German fest), from PIE root *past- "firm" (source of Sanskrit pastyam "dwelling place").

The adverb meaning "quickly, swiftly" was perhaps in Old English, or from Old Norse fast, either way developing from the sense of "firmly, strongly, vigorously" (to run hard means the same as to run fast; also compare fast asleep), or perhaps from the notion of a runner who "sticks" close to whatever he is chasing.

The sense of "living an unrestrained life" (usually of women) is from 1746 (fast living is from 1745). Fast buck recorded from 1947; fast food is first attested 1951. Fast-forward first recorded 1948. Fast lane is by 1966; the fast track originally was in horse-racing (1934); figurative sense by 1960s. To fast talk someone (v.) is recorded by 1946.
fast (v.) Look up fast at Dictionary.com
Old English fæstan "to fast" (as a religious duty), from Proto-Germanic *fastejan (cognates: Old Frisian festia, Old High German fasten, German fasten, Old Norse fasta), from the same root as fast (adj.).

The original meaning was "hold firmly," and the sense evolution is via "firm control of oneself," to "holding to observance" (compare Gothic fastan "to keep, observe," also "to fast"). Presumably the whole group is a Germanic translation of Medieval Latin observare "to fast." Related: Fasted; fasting.
fast (n.) Look up fast at Dictionary.com
Old English fæstan, festen, or Old Norse fasta; from the root of fast (v.).