Through training. Just like in the original Naruto series, characters with multiple bloodlines would need to train hard to master their abilities. They might start with basic techniques related to each bloodline and gradually build up to more complex ones.
Some characters could develop their powers through unique experiences. For instance, if a character has a bloodline related to water and another related to lightning, being in a situation where they are surrounded by water during a lightning storm might trigger a new power or a way to combine those two bloodlines in a new way.
In such fanfictions, the powers from multiple bloodlines in Naruto are often shown in an exaggerated way. For example, if a character has Uchiha and Uzumaki bloodlines, they might be able to combine the Sharingan's visual prowess with the Uzumaki's chakra reserves and sealing abilities. It makes for really interesting and unique combat scenarios.
Some characters might develop their gambit powers through intense training. Just like in the original Naruto series where characters train hard to master their ninjutsu, in fanfiction, they could push their limits in special training regimens to unlock new aspects of their gambit powers.
Often, Naruto Shiki's powers develop through intense training. He might face difficult challenges that force him to dig deep and discover new aspects of his abilities. For instance, in a fight against a powerful enemy, he realizes he can merge his shadow powers with his existing ninja techniques to create a more powerful attack.
In the 'naruto has 3 bloodlines' fanfiction, these bloodlines can have a profound impact on the story. For instance, if one of the bloodlines allows Naruto to communicate with animals in a special way, it could lead to new alliances and ways of gathering information. A bloodline that enhances his ninjutsu in a unique way might make him the target of other villages or rogue ninjas who want to steal his power. And a bloodline that gives him the ability to heal others more effectively could change his role within the village from just a fighter to also a healer, which would completely shift the relationships he has with other characters and the overall direction of the story.
The relationships can be developed in many ways. For example, if a character has the Uchiha and Uzumaki bloodlines, other characters might be initially wary due to the power of the Uchiha but also drawn to the kindness associated with the Uzumaki. Naruto, if he had multiple bloodlines, could use his different abilities to help his friends in new ways, strengthening their bonds. For instance, using a bloodline - specific healing ability to save a wounded friend.
They can develop through the new combat skills they gain from their multiple summons. Different summons bring different abilities to the table. A character might learn to use the speed of one summon and the strength of another in combination, thus evolving their fighting style. Also, dealing with the different personalities of the summons can teach a character patience and leadership.
The characters can develop differently in terms of their relationships. For example, Naruto might have closer ties with some tenants due to shared interests or goals. He could learn different jutsu from them and thus have a more diverse skill set.
Another concept is the origin of these multiple bloodlines. In fanfiction, some might explore the idea that there was an ancient clan that had all these bloodlines but they were split over time. This could lead to a story where a character discovers their heritage and has to learn to control all these different powers.
One interesting combination could be the Uchiha bloodline with the Uzumaki bloodline. The Sharingan from the Uchiha and the massive chakra reserves and sealing abilities of the Uzumaki could create a really powerful Naruto. Another could be the Hyuga bloodline combined with the Uzumaki. The Byakugan's visual prowess along with the Uzumaki's chakra would give Naruto enhanced perception and more chakra to use for powerful jutsu.
They often start with a unique event. For example, Naruto could be exposed to a powerful chakra source that changes his physiology. This new source of chakra could be something ancient and otherworldly. Another way is through a bloodline awakening. Maybe Naruto discovers he has a hidden bloodline that gives him godlike powers. This could be related to a long - lost clan in the Naruto world.