On-screen, it has never been made clear how the ongoing enmity between the Time Lords and the Daleks developed into full-scale war.
Off-screen, Russell T Davies and others involved in the writing have stated that, in their opinion, the "opening volley" of the Time War was the events of "Genesis of the Daleks", when the Time Lords assigned the Fourth Doctor to prevent the Daleks' creation. An even earlier story, "The Evil of the Daleks", has also been cited as a potential early flashpoint & it's possible to find signs dating all the way back to the Doctor's first encounter with them in "The Daleks".
The expanded franchise legitimized the above in the audio series Gallifrey, though it suggests the Daleks may have fired the first shot.
In the Gallifrey series 3 episode "Panacea" (audio story), the Daleks unleashed the Dogma Virus upon the Time Lords of Gallifrey, which led to a chain of events during which Romana (now Lord President of Gallifrey) was forced to flee the planet and to spend years on another Gallifrey in an alternate timeline.
In the Gallifrey series 6 episode "Ascension" (audio story), the Dogma Virus was eliminated but for a time it was thought that Romana had been lost. Because of this, a grief-stricken Coordinator Narvin -- a colleague who is depicted as having fallen in love with Romana II -- sought revenge against the Daleks & ordered that the Fourth Doctor be assigned to wipe the Daleks from history, leading to the events of "Genesis of the Daleks". (After doing this, Narvin learned that Romana was not lost after all.)
It is strongly indicated in dialogue that this attempt to wipe them from history will lead to a war with the Daleks. However, it is not clear that this is the Last Great Time War spoken of on TV. The BBC Books novels also chronicle a Time War between the Daleks and Time Lords (during which Romana II regenerated into Romana III), and Russell T Davies has stated that this is a separate event from the Last Great Time War. (In real life, the events of World War II were very much seeded by the events of World War I & a similar connection is possible between the two Time Wars.)
Davies, for his part, suggested that it was important to keep the Time War a mystery, though the recent episodes "The Night of the Doctor" and "The Day of the Doctor" have removed some of this mystery.