So much is made of the myth of Kore becoming Persephone that I know very little about Demeter beyond that myth. Below is some interesting information that I found on Theoi.com.
Demeter was the goddess of all the earth’s fruits, not only grain (though this was by far the most important food-crop). As the goddess of horticulture, the growing of figs, apples, pears and other fruits fell within her dominion (with the exception of olives, the province of Athena, grapes, the province of Dionysos, and pomegranates). She was also the goddess of vegetable crops (with the exception of beans, which for some reason were regarded as impure).
Demeter was depicted in scenes of the Gigantomachia (War of the Giants) wielding a sword, which may have been the golden blade; or the blade might instead be a golden sickle which she used to harvest the first crops of wheat. Some say she found the scythe that Kronos used to castrate his father Ouranos on the island of Sicily, and employed this in the first harvest of the grain. Demeter was called Khrysaoros (Lady of the Golden Blade) after the golden sword or sickle which she wielded. Demeter’s chariot-car was drawn by a pair of winged Drakones (serpent-like beasts). The goddess also employed these monsters as guards and attendants.
Demeter carried a pair of flaming torches in her search for Persephone. She was often depicted in art holding one or two torches as her attribute.
The serpent (a creature which represented rebirth in nature and the fertility of the earth) was the animal most sacred to Demeter. A pair of winged-serpents drew her chariot. The gecko, found buried under rocks, was like its relative the snake, regarded as sacred to the goddess Demeter.
The pig was an animal offered to Demeter in sacrifices ensuring the fertility of the earth.
The turtle-dove was a bird sacred to the goddess Demeter (as well as Aphrodite).
The red-mullet was a fish regarded sacred to Demeter in the cult of the Eleusinian Mysteries.
As the goddess of agriculture, wheat was Demeter’s sacred plant. Barley was also sacred to agricultural Demeter. The plant was used with mint and honey in a sacred drink of the Mysteries.
The poppy which grew wild amongst the grain-fields was regarded as sacred to Demeter, and was worn by her priestesses.
The first fruits of the season were offered to the gods of farm and field:
Demeter received the first corn, Dionysos the first grapes, and Athene the first olives. The Thesmophoria festival of Athens was one such festival celebrating the offering of the first fruits. Similar rites were practiced throughout Greece. More information on these rituals can be found in the various regional entries below.
Presumably sneezes were attributed to Demeter because they were associated with harvest (hay-fever) and milling (from flour dust).
Aelian, Historical Miscellany 12. 31 (trans. Wilson) (Greek rhetorician C2nd to 3rd A.D.) :
“Let me give you the names of Greek winds that were taken seriously by the ancients. They had one called Pramnian, which was apparently sacred to Demeter.”
Demeter means Earth Mother; Deo means of the earth.
Epithets: Aglaokarpos (Giver of Goodly Fruit), Aglaodôros (Bestower of Splendid Gifts), Amphiktyonis (of the meeting place?), Anesidora (Sender-up of Gifts), Anassa (Queen/Lady), Antaea (the approachable?), Brimo (the angry/terrifying), Despoina (The Mistress), Dia Thea (Bright Goddess), Epogmie (Of the Furrows), Erinys (Fury), Eukomos (Lovely Haired), Eunostos (of mills), Europa (of Europe), Eustephanos (Lovely Crowned), Hagne (Pure), Herkyna (Of the Stone Enclosure), Horephorus (Bringer of Seasons), Kabeiraie (of the Kabeiroi), Kallisphyros (Beautiful Ankled), Kallistephanos (Beautiful Crowned), Karpophoros (Fruitbearer), Khloe (Green), Khrysaoros
(Lady of the Golden Blade) , Khthonia (of the Earth), Kidaria (of royal head-dress), Kyanopeplos (Dark Veiled), Kydrê Thea (Noble Goddess), Lousie (Bather), Malophoros (Bearer of Fruit), Megala Thea (Great Goddess), Megala Mêter (Great Mother), Melaina (Black), Panakhaia (of all the Greeks), Ploutodoteira (giver of wealth), Polyphorbos (All Nourishing), Pron (of the Headland), Prostasia (Patron-Goddess or Protector), Potnia (Queen), Potnia Theaon (Queen amongst Goddesses), Pygaiê (of the Gates), Rhea eukomos thugater (Daughter of Rich-haired Rhea), Semne (Holy), Sitos (Giver of Food/Corn), Thermasia (Warmth), Thesmia (of the Laws), Thesmophoros (Lawgiver), Xanthê (Blonde).