In Taming of the Shrew, identify and explain some classical allusions from Greek and Roman mythology in Act 1 Scene ii.
The following five gods/goddesses are mentioned in the text you cited.
Adonis - he was the lover of Aphrodite. It was said that when he died, Aphrodite (Venus in Roman mythology) rushed to his side and sprinkled his blood with blood. He was then reborn into a flower called the anemone.
Cytherea - Another name for Aphrodite
Io - a nymph that was a priestess to Hera and an constant object of affection in Zeus's eyes. She is one of the many females that Zeus cheats on Hera with. In the most famous myth about her, Zeus transforms here into a white bull and himself into a cloud to disguise his infidelity from Hera, but ultimately, she figures out his tricks (check out Wikipedia for more detailed info).
Daphne & Apollo: Once Apollo (god of music, light, etc.) was struck by an arrow by Eros, he became infatuated with the nymph Daphne. Apollo chased after Daphne to the point where she turned herself in to a laurel (a type of plant with branches and leaves) to hide from him.
It looks like you've got some allusions here to stories from the Classical Greek about love, infatuation, and flirtation. Each of these stories prelude the love, infatuation, and flirtation that is about to take place in the play The Taming of the Shrew.
Many of the references in this section derive from Ovid's Metamorphoses, a work that would have been familiar to all educated people in Shakespeare's period.
Apollo: Apollo was a Greek god of the sun. He is also associated with prophecy, music and healing. He was son of Zeus and Leto and brother of Artemis. He is often depicted playing a lyre.
Nightingales: In Greek mythology, there were two sisters, Philomela and Procne. Tereus married Procne but later raped her sister Philomela and cut out Philomela's tongue. In revenge, Procne killed their son Itys, chopped up his body, and fed him to Tereus for dinner. The gods transformed Tereus into a hawk and Procne and Philomela into a swallow and a nightingale (sources differ as to which sister became which bird). The song of the nightingale was thus considered extremely mournful.
Semiramis is mentioned in Greek sources as a semi-mythological queen of Assyria, descended from a fish goddess.
Adonis was a mortal youth who was extremely handsome and loved by Aphrodite. When he died, he was transformed into an anemone.
Cytherea is another name for Aphrodite or Venus, the ancient goddess of love.