Here's a list of common Greek roots and their meanings which I compiled myself. Any questions, comments, corrigenda, or addenda, don't hesitate to msg me.
A note on the spelling: All these roots are spelled in the Latin way, which means C for kappa, the diphthongs AE for alpha-iota and OE for omicron-iota/omega-iota, Y for upsilon, and N for gamma-nasal. In English they might be spelled with K for kappa or simple E for either diphthong. Thus the Greek "oikia" (οικια) gives the Latin "oeconomia" and the English "economy" (formerly "oeconomy"). You see how it works.
Root Meaning As In
a(n) not agnostic
acou hearing acoustic
acr high acrophobia
adelph brother Philadelphia
aer air aerodynamic
aesthes, -t perception anaesthesia
aeti cause aetiology
agath good agathodemon
agog guide pedagogue
agon contest protagonist
agora marketplace agoraphobia
agr field agronomy
all other allophone
amph both amphitheatre
ana up anabolic
andr male androgenous
angel messenger angel
angi vessel angiosperm
anth flower chrysanthemum
anthrop mankind anthropology
anti against antibody
apo away apogee
arch(ae) beginning OR ruler archaeology, monarch
arist best aristocracy
arithm number arithmetic
aroma smell aroma
arthr joint arthroscopy
ast(e)r star asterisk
athl athlete athlete
aut self automobile
ball see bol
bapt dip baptist
bar heavy baritone
bas base OR walking basis
bathy deep bathysphere
bi(o) life biosphere
bibli book bibliophile
bol, ball cast ballistic
brach(y) short tribrach
cac bad cacophony
calyps, -t covered calypso
card heart tachycardia
carp fruit mesocarp
cata down catastrophe
centr center centre
cephal head cephalization
ch(e)ir hand sthenochire
chiasm cross chiasma
chlor green chlorine
chord string chordophone
chore dance choreography
chrom colour chromatic
chron time chronoscope
chthon ground chthonian
chym juice chyme
cin motion cinema
clin tilt clinic
coel hollow coelom
cosm(et) world OR ordered cosmopolitan, cosmetic
crac, crat power democrat
cras mixing crasis
cris, crit sift, discern criterion
crypt hidden cryptography
cyan blue cyanogen
cycl circular cyclic
dec ten decathlon
dem populus democracy
dendr tree dendrite
derm skin dermatologist
di two dilemma
dia through diagnosis
dox opinion orthodox
drom racecourse dromedary
dynam power dynamo
ec(to), ex out ectothermic
electr amber OR electricity electrum, electric
en(do) in endogenous
epi upon epitaph
epistem knowledge epistemology
epistol letter epistolary
eros, -t sex erotic
erg, urg work energy
erythr red erythrocyte
ethn race ethnic
eu well euphonium
eury broad eurypterid
gam marriage monogamy
ge earth geography
gen(es), -t birth OR type genetic
gloss, glot tongue glossary, polyglot
glyph carving hieroglyph
gnath jaw gnathometer
gnom judgement OR aphorism gnomic
gnomon sundial's index gnomonic
gnos know gnostic
gon offspring OR angle gonad, polygon
gram written message telegram
graph write telegraph
gymn naked gymnasium
gyn(aec) woman gynecology
gyr circular motion gyroscope
haem blood hemoglobin
heli sun helium
hellen Greek hellenophile
helminth worm platyhelminth
hemer day ephemeral
hept seven heptad
hex six hexad
hier temple hierophant
hipp horse hippopotamus
hydr water hydration
hygr moisture hygroscope
hyper over hyperbaric
hypn sleep hypnotic
hypo under hypochondriac
ichthy fish ichthyology
iod violet iodine
iso equal isometric
itis irritation conjunctivitis
labyrinth maze labyrinth
leuc white leucocyte
lith stone lithography
log word OR study dialogue
lys, -z, -t break OR loosen analyze
macr long macron
math learning math
meg(al) big megalomania
melan black melanin
mer part polymer
meta with OR after metamorphic
met(e)r measure meter
micr small micron
mim imitate pantomime
mis hatred misogynist
mnem, -s, -t remember mnemonic
mon single monad
mor fool sohpomore
morph form morphology
music music music
myc fungus mycology
narc numbness narcotic
ne(o) new neoplast
necr corpse necrophilia
neur nerve neuron
nom law nomic
ochr brown ochre
oct eight octagon
odon(t) tooth pterodon
oec home ecology
oen wine oenology
olig few oligarchy
onc swelling oncogene
onom(at), onym name pseudonym
o(ö) egg oocyte
ophthalm eye ophthalmoscope
ops, opt(ic) sight optics
orex appetite anorexia
ornith bird ornithology
orth correct OR straight orthography, orthorhombic
paed child pedophile
palae ancient palaeography
pan(t) all panorama
para alongside paranormal
parthen virgin parthenogenesis
path disease pathology
pent five pentagram
peri around periscope
phae gray phaeochrome
phaen appear phenomenon
phag eat xylophagous
phil love philanthropy
phob fear homophobia
phon sound phonic
phot light photography
phyll leaf chlorophyll
phyt plant bathyphyte
pin drink pinocytosis
plast mould rhinoplasty
platy flat platyhelminth
pless,
plex, plect strike apoplexy
pleth multitude plethora
pnoe, pneum(at) breath pneumonia
pod foot gastropod
poeia, -sis, -t do, make OR poem poet
polem war polemic
polis, -t city metropolis
poly many polygamy
pragm action pragmatic
prot first protozoa
pseph voting psephology
pseud false pseudonym
psych soul psychology
pyr fire pyre
rhabd rod rhabdomancy
rhe, rrhoe flow diarrhea
rhem(at) word rhematic
rhin nose rhinocerous
rhiz root mycorrhiza
sarc flesh sarcasm
scop view endoscopy
sem symbol semantics
som body somatic
soph wisdom sophistry
sperm seed sperm
sphaer sphere ionosphere
spongi sponge sponge
spor sowing spore
sten narrow stenography
ster(e) solid OR 3D stereo
sthen strength sthenochire
stigm(at) mark OR puncture stigmata
stom mouth colostomy
strat(eg) army strategy
strob whirl stroboscope
stroph, -eph,
-eps, -ept turn OR twist catastrophe
syn, sym with OR together symbiotic
tach(y) fast tachyon
taph tomb cenotaph
techn craft technical
tect builder architect
tel(e) far telephone
tetr four tetrad
thalass, -tt ocean panthalassa
thanas, -t death euthanasia
thaum(at) miracle thaumaturgy
the god theocracy
theatr theatre theatre
therap therapy therapy
ther(i) beast megatherium
therm heat thermometer
thes, thet position thesis
tom cut hysterectomy
tox(ic) poison OR arrow toxic
tri three trigonometry
troph nourish atrophy
urg see erg
xanth yellow xanthochrome
xen strange xenophobe
xer dry xerox
xiph(i) sword xiphoid
xyl wood xylophone
zo(ö) animal zoo
zyg yoke OR connection zygote
zym fermentation zymology