What is the relationship between Apollo and the Furies? How do they view each other?

Their relationship is antagonistic, to say the least. Whereas Apollo is sympathetic to Orestes for killing Clytemnestra and Aegisthus—after all, he did tell Orestes to kill them—the Furies want to exact a terrible vengeance on him. Later on, Apollo intervenes to help Orestes escape to Athens while the Furies are asleep. He tells him that once he's reached Athens, he should seek the assistance of the goddess Athena.
As one can imagine, the Furies are suitably furious when they find out they've been tricked by Apollo and on waking up, immediately set off to hunt down Orestes. Unfortunately for them, Orestes is under Athena's protection. His fate will be determined by a formal trial at which Apollo will speak on his behalf and the Furies will act as his prosecutors. As it has been set up, the trial neatly encapsulates the mutually antagonistic relationship between the two parties.
To the Furies, Apollo represents an attack on justice, on the ancient laws, and on the way things have always been. They see his defense of Orestes as an attempted power grab from the older gods such as themselves. For his part, Apollo accuses the Furies of dishonoring the institution of marriage as established by Zeus himself. In their condemnation of Orestes, he argues, they overlook the nature of marriage as a sacred oath, one that in this case was violated by Clytemnestra when she conspired to have Agamemnon killed.

The Oresteia is one of many Greek literary works that takes places immediately after the Trojan War. Both the plot of the story and the characters within it would have been familiar to its audience.
The most important information you need to understand the relationship of Apollo to the Erinyes or Furies is actually the background found in Hesiod's Theogony about the generations of the Greek gods. 
According to Hesiod and other ancient sources, the first generation of gods were primordial deities representing such natural elements as Chaos (or the Void), earth, night, and sky. The second generation sprang from these and overthrew them; this second generation was that of the Titans. The Furies were part of this generation. The third generation of Olympian gods (including Apollo) overthrew the Titans, imprisoning many of them or condemning them to eternal punishment.
The Furies are female goddesses of justice who see Apollo's support for Orestes as part of a pattern of the disrespect of the upstart younger gods for an older generation. They also argue that Apollo as a male god is advocating the rights of men in preference to those of women and mothers. Apollo sees the Furies as primordial, irrational, and vengeful. Thus the relationship is antagonistic until Athena, a female goddess of wisdom, creates a brilliant compromise, founding the Court of the Areopagus.