The baby might inherit Naruto's ability to use chakra in large amounts and his signature Rasengan. Since Naruto has such a vast amount of chakra, it's likely that his child would also have a significant chakra reserve.
The training could begin as a form of self - defense. Given Naruto's history of facing many enemies, he would want his child to be able to protect itself. He and Hinata might start by teaching the baby how to sense danger. They could use games, like hiding and seeking with a 'dangerous' element added. For example, Naruto would create a harmless clone that the baby has to detect and avoid. This would help the baby develop both its sensing abilities and its reflexes.
Perhaps the baby could be named Kazuki. It gives a sense of being strong and noble, which would be fitting for the child of Naruto and Hinata. In fanfiction, the name could also be symbolic of a new beginning or a special power that the baby might possess, like being related to the combination of Naruto's Nine - Tailed Fox power and Hinata's Byakugan abilities.
Sakura would be overjoyed. She's been Naruto's friend for a long time, and she would be excited to see him start a family. She might offer to be the baby's doctor and make sure the baby is healthy. Shikamaru would probably be a bit lazy about it at first but would come around and offer his strategic mind to help protect the baby if needed.
Naruto would likely become more responsible. He's used to being the hero, but now he has to be a father too. So he might cut back on some of his risky adventures to be there for his family. Hinata would become more confident as a mother. She's always been a bit shy, but with a baby to protect, she'd find her inner strength.
The baby might inherit Naruto's ability to use the Nine - Tails' chakra. Since Naruto has a strong connection with the Nine - Tails, it's possible for the baby to have some form of access to that power. This could manifest as enhanced strength and speed, similar to when Naruto taps into the Nine - Tails' chakra.
In fanfiction, if Naruto and Yoruichi had a baby, the baby could potentially have a wide range of powers. Naruto is known for his vast chakra reserves and his ability to use shadow clones and the Rasengan. Yoruichi, on the other hand, is a master of Shunpo and has cat - like abilities. The baby might inherit Naruto's chakra control and Yoruichi's speed, creating a unique combination. For example, it could use chakra - enhanced speed to quickly move around the battlefield and then deliver a powerful Rasengan - like attack with a touch of Shunpo - style finesse.
Hinata's family would also have to deal with the emotional aspect. They love Hinata in her normal state, and now they have to adjust to this new form. They might start to remember when Hinata was actually a baby before and all the hopes and dreams they had for her then. Some of the more conservative members of the family might be more hesitant to let Naruto be around her, but the more open - minded ones could see that Naruto's presence is actually good for her in this state. And they would all be united in their goal to get Hinata back to her normal self.
The baby could have Naruto's ability to use the Rasengan. Since Naruto has mastered this powerful technique, it might be passed down genetically or through training. Also, it could inherit Samui's intelligence and strategic thinking. This combination would make the baby a very formidable ninja.
Their relationship would likely become even more intimate. Naruto would be more tender and caring towards Hinata. He'd be constantly worried about her well - being.
Their friends would be overjoyed. Sakura would probably be really excited and offer to help Hinata with any medical advice during the pregnancy. Shikamaru might give Naruto some practical advice on how to handle the added responsibilities. Kiba could be really happy and joke around about the new baby being a future ninja partner for his own pups. And Ino would be eager to plan a baby shower for Hinata, full of beautiful flowers and gifts.