The ancient Hellenes were not fearful of snakes. They might have been cautious of poisonous ones, but in general, happening across a snake was a good omen. Unlike in Jewish and Christian mythology, where the Devil working though a snake got Eve to eat the apple, Hellenic mythology usually reserves a very positive place for snakes. Today, I'm giving some examples of the positive images surrounding serpents and snakes, although there are, surely, also negative ones to take into account.
Asklēpiós was, and is, a much beloved Theos. He started out being honored as a hero--the son of Apollon and Koronis--but became a God in His own right because of his healing skill. Worship places of Asklēpiós were called 'asklepieia' (Ἀσκληπίεια). An asklepieion (Ἀσκληπιεῖον) served as a temple, a hospital, and as a training-institute of the healing arts. In ancient Hellas, the sick would come to an asklepieion and offer a sacrifice to Asklēpiós--amongst the recorded sacrifices are black goats or sheep, gold, silver, or marble sculptures of the body part that required healing, and coins--in hopes of healing. They would then settle into the abaton (άβατον) or enkoimeterion (εγκοιμητήριοn), a restricted sleeping hall, which was occupied by the sick alone, or sometimes by a group of them, as well as a good few snakes, which are considered sacred animals of Asklēpiós.

In mythology, after His training is complete, Asklēpiós receives the blood of Médousa from Athena. Drawn from two different blood vessels in Médousa's neck, some of it can kill, and some of it can heal even the dead. Asklēpiós uses the blood to resurrect the dead, but this is against the wishes of Zeus, who kills Him. He is either placed amongst the stars as the constellation Ophiuchus, the Serpent Holder, or revived by Zeus as a God to satisfy a furious Apollon.

The Drakones were named after the Greek 'drakein' and 'derkomai, meaning 'to see clearly' or 'gaze sharply'. These were guardians, usually of wells and springs, groves, Gods, or treasure. As guardians, they were usually equipped with sharp fangs, deadly poison and/or multiple heads. In essence, they were however seen as giant snakes which--and this is wholly a personal observation--makes sense when most protective and purifying Theoi were depicted as snakes.

All sources but the ones where Agathós Daímōn is identified as Theos, represent the Agathós Daímōn as a snake; this applies to both artwork as assumed physical appearance. The Agathós Daímōn was always a positive in one's life, and was generally seen as the source of personal or familial good fortune. Libations of (unmixed) wine were given to Him with each newly opened case of wine, and during feasts and symposium, Agathós Daímōn received the first libation. When crossing a snake on the road, it was also customary to pour out a libation, just in case it was a herald of Agathós Daímōn, or Agathós Daímōn Himself.

It should be said that Harrison believes Zeus Meilichios is an epithet of Zeus superimposed over an existing snake God: Meilichios, a 'home-grown, autochtonous [deity, from] before the formulation of Zeus'. Even more telling: the cult of Meilichios was very pronounced in Boeotia, where He was worshipped as a provider of wealth (Harrison, p. 21). I pose that, at the same time Zeus became equated with Meilichios, so did the Agathós Daímōn; a daímōn of good fortune (most likely through fertility and good harvest, the two greatest blessings from the Theoi), superimposed over the snake God Meilichios, exchanging positive qualities while assuming immortality for Himself. Zeus Meilichios adopted Meilichios' cleansing and purifying qualities.

In some cult worship, Agathós Daímōn was a male deity, who was married to the Theia (daímōn?) Agathe Tyche. Their worship was known in Athens, and They had a temple at Lebadeia, in Boeotia, where one could visit the oracle of Trophonios--but only after spending a fixed number of days in a building, which was sacred to the 'Agathei Theoi'--which probably refers to Agathe Tyche and Agathós Daímōn together--and most likely housed one or several snake(s). It was this building the suplicant was brought back to when he returned from the oracle--usually passed out from the experience--in order to recover.

Archaeologists suspect that Athena, Médousa and Poseidon found their origins in Libya. They came to Hellas through Crete at the dawn of Hellas. In the beginning of Her rein, Athena may have been a snake and fertility Goddess--a trait she shared with her Libyan counterpart, who had Her own cult--and may have either had a priestess who fit the Médousa myth or--and this is more likely--Médousa had her own cult as a snake, fertility and (menstrual) blood Goddess. Especially the latter may be linked to the myths concerning Médousa's blood.

Athena's role as a snake and fertility Goddess is still visible in the myth about the child she had with Hēphaistos;  Erichthonios (Ἐριχθόνιος), who was half man, half snake. It's even posed that in the early days, Athena was married to Hēphaistos and had His child willingly. As Athena was stripped of Her roles as a fertility and snake Goddess, Médousa's myth came into being, where Athena distances Herself from sex and snakes, by punishing an epithet of herself (Athena Tritogeneia, perhaps: 'born of Trito', a lake which was supposedly located in Libya), or the Libyan snake Goddess Médousa, who may have still been attached to Her worship. By placing Médousa's head on Her breastplate or shield, Athena's mythology is continuously linked to Her Libyan heritage, but harmlessly so, to Her new image of a virginal warrior.

Few references remain to Médousa's Libyan cult. There's vague reference to Médousa being a patron of Libya as a whole, or that she was the Goddess most worshipped by the Amazons. She was linked to protection, snakes, menstrual blood, blood, fertility, and femininity in general. If this is true, it's understandable why her worship did not match the Hellenic religion: for one, she's most likely a very powerful female deity. This did not match the hierarchy of the ancient Hellens, and so, Médousa became a monster, and was dealt with accordingly. Blood was one of the fluids that caused serious miasma, and menstrual fluid wasn't even spoken of in ancient Hellas, let alone revered. Not a single Goddess would have it in their portfolio.
Of course, this is not a complete list, but it is a starting point, and one I might build further on. Now our snake season is about to start again (and my girlfriend is fearfully eying the pond in our garden) I figured it would be good to have a reminder of the important mythology and customs surrounding snakes.