Naruto's view of family would be different. Instead of longing for a family like in the original story, he would have a real family with Sarutobi. This would change his motivation. Instead of wanting to be Hokage to gain recognition, he might want to be Hokage to continue Sarutobi's legacy and protect the family he now has in the village.
Another plot development could be that with Sarutobi as his adoptive father, Naruto has access to different mentors. Sarutobi could call in favors from other great ninjas to train Naruto. This would expose Naruto to a wider variety of fighting styles and strategies. Maybe he gets trained by Kakashi earlier, or even has the chance to learn from some of the retired or older ninjas who usually keep to themselves.
Perhaps the plot could involve Sarutobi protecting Naruto from the more malicious elements in the village. He could use his influence to ensure Naruto has a normal childhood, or as normal as it can be for a jinchuriki. Naruto might also inherit some of Sarutobi's wisdom and leadership skills, which would be important for his future development.
Naruto would likely be more confident. With Sarutobi as his adopted father, he would have a strong support system. He might not be as brash as in the original story, as Sarutobi's teachings would instill more discipline in him.
One possible plotline could be Naruto growing up with a different set of values and training methods. Instead of the rough start he had in the original story, he'd be mentored by the Sarutobi clan's elders. He might learn more about the history and secrets of Konoha at an earlier age, which could lead to him being more politically aware and better equipped to handle future threats. For example, he could be trained in unique Sarutobi jutsu from a young age, giving him an edge in combat.
His character would likely be more refined. With the Sarutobi clan's influence, he'd have better access to knowledge and proper guidance. He might not be as brash as in the original story. For example, he'd learn the importance of diplomacy and strategy earlier.
In this fanfiction scenario, Naruto might learn unique chakra control techniques from Sarutobi. Sarutobi, being an experienced ninja, could teach Naruto how to better manage his large chakra reserves. This would lead to Naruto being able to perform jutsus more efficiently and with less waste of chakra.
I really like 'Sarutobi's Unseen Trials'. In this fanfiction, it focuses on the political aspects that Sarutobi had to deal with in the Hidden Leaf Village. It gives a new perspective on his leadership and the difficult decisions he had to make.
Perhaps in this fanfiction, Sarutobi is trapped in a genjutsu created by a rogue ninja. Naruto, being highly resistant to genjutsu due to his Uzumaki blood, manages to break into the genjutsu world. There, he has to fight his own inner demons as well as the rogue ninja's illusions to free Sarutobi.
Obviously Naruto would be one of the main characters interacting with him. Naruto would look up to Sarutobi as a mentor and a source of knowledge. Their interactions could range from training sessions to deep conversations about the future of the village.