There are 3 total Ninjutsu in the game, all of which can be acquired on the first playthrough, and are obtained by defeating certain bosses. They are as follows:
Bloodsmoke Ninjutsu: Obtained through the main story. You will obtain this after defeating Genichiro Ashina for the first time at Ashina Castle.
Puppeteer Ninjutsu: Also obtained through the main story. You will obtain this after defeating the Folding Screen Monkeys at Senpou Temple, Mt. Kongo.
Bestowal Ninjutsu: This is the only Ninjutsu that is not given to you through the main story. To obtain it, you must first defeat the Guardian Ape in the Sunken Valley. Then, you must progress to the Ashina Depths and again fight the Guardian Ape (now the Headless Ape). After defeating him, you will acquire it.