Thetis was a nymph whom both Zeus and Poseidon wished to marry, but a prophecy revealed that the son of Thetis would be greater than the father, so both Poseidon and Zeus stepped down in favor of a noble human, King Peleus. Thetis was awarded Peleus by Zeus for noble behavior, but Thetis was displeased with having to marry a mortal. Artistic depictions of the wooing show Peleus clinging to a shapeshifter. Peleus proves up to the challenge and they wed. The marriage of Thetis and Peleus was a grand affair on Mt. Pelion, with all the gods and goddesses. Unfortunately, the list of guests had one important omission, Eris, goddess of discord. In response to the slight, she gave a gift of a golden apple to the most beautiful of the goddesses. This led to the Judgment of Paris, abduction of Helen, and the Trojan War.
As to the maternal behavior of Thetis... after her attempts to immortalize her infant, as per this bath, dipping him in the River Styx, or burning off his mortality, were interrupted, Thetis took off in a huff*, leaving Achilles in the care of his father.
Peleus took the instructional course most popular for young heroes. He farmed him out to the centaur Chiron for fostering.
*In some accounts, Thetis and Peleus live together during Achilles' upbringing. Thus, Thetis is there to see Achilles off to war.
