The Time War was first explicitly shown on screen in "The End of Time" during which we saw Gallifrey prior to its apparent destruction. This was next followed by the minisode "The Night of the Doctor" which takes place during the war and reveals the Doctor's evolution into a warrior. This was followed by another minisode, "The Last Day", set during the final day of the Time War, but not featuring any Doctor.
"The Day of the Doctor", the 50th anniversary special, revealed that the Doctor's ninth incarnation, known as the War Doctor, was the one that fought in the war. It also revealed that the loss of Gallifrey and final destruction of the Daleks unfolded differently than history (and the War, Ninth, Tenth and Eleventh) Doctors remembered. Technically all Doctors - including the Twelfth - participated in saving Gallifrey and destroying the Daleks and ending the Time War.
Prior to these events, there had been references to the Time War and the apparent destructuin of Gallifrey starting with the Ninth Doctor story "The End of the World", with subsequent references in episodes such as "Dalek", "Bad Wolf", "The Parting of the Ways", "Doomsday", "Gridlock", "Utopia", "The Sound of Drums", "The Stolen Earth", "Journey's End", "The Doctor's Wife" and "The Name of the Doctor".
Writers and producers involved with the series, such as Russell T Davies, have suggested that earlier stories such as the Fourth Doctor story "Genesis of the Daleks", the Seventh Doctor story "Remembrance of the Daleks" (in which the Doctor supposedly destroys Gallifrey) and even the Second Doctor story "The Evil of the Daleks" were all "early volleys" in the Time War. The Big Finish audio drama Gallifrey: Ascension suggests the events of that spin-off series indirectly put in motion events leading to the Time War.
As for depicting the "final destruction" of the Daleks, pretty much every classic-era story played with that trope. However the stories that are generally seen as suggesting a final end (though of course the Daleks can never been totally exterminated) include "The Daleks' Master Plan", "The Evil of the Daleks" (which was intended to write the Daleks out of the series for good), "Remembrance of the Daleks", "Dalek" and "The Parting of the Ways".