Watch the eight seasons in the order they aired, from the pilot on 10 September 2009 to «I Was Feeling Epic» on 10 March 2017. There are no specials or films to slot in between.
The spin-off The Originals grows out of season 4: its backdoor pilot is episode 20 of that season, titled «The Originals». If you want both shows, start The Originals after finishing season 4 of The Vampire Diaries and run them side by side.
Legacies, the second spin-off, follows the end of The Originals and is set at the Salvatore School, so it is best left until both earlier series are done.