Well, Sirius might feel a bit protective of Harry. He could pull Harry aside and have a serious talk with him about being responsible and making sure this relationship doesn't interfere with their fight against the dark forces. He might also warn Ginny to look out for Harry.
One possibility is that Sirius could be initially shocked. But then, being the understanding person he is, he might give them his blessing. After all, he has seen how Harry has grown and Ginny is a great girl.
It's also possible that Sirius could be a bit amused. He has always been a bit of a rebel himself. He might tease them a little bit, but in a friendly way. Then he could offer some advice based on his own experiences with relationships, if he has any.
One possibility is that Harry would be extremely hurt and angry. He might immediately confront Ginny about it. He could cut off all ties with her and distance himself from their mutual friends as well, feeling betrayed not only by Ginny but also by the false image he had of their relationship.
Perhaps Snape would force them to work together on a task as punishment. This could be something like cleaning the Potions classroom without magic or helping in the Forbidden Forest. During this time, they might start to understand each other better. Harry could see that Draco isn't just a spoiled brat all the time, and Draco could see that Harry has more depth than he initially thought.
Well, in that situation, Sirius would be shocked and heartbroken. He might first scold Harry for doing something so dangerous to himself. Then, being the impulsive person he can be, Sirius would probably take Harry to see someone he trusts, like Dumbledore or Madam Pomfrey. He would not let Harry out of his sight until he was sure Harry was okay and not likely to do it again.
In fanfiction, when Harry and Ginny meet Sirius, Sirius might offer them some sage advice on dealing with the dark forces. He could share his experiences about fighting Voldemort and his Death Eaters. Maybe he would see the potential in Ginny and encourage her to further develop her magical abilities, as he has always been supportive of Harry.
Another scenario could be that Ginny accepts but on the condition that they go on an adventure first. They could travel around the wizarding world, visiting different magical places they've only heard about, like the hidden valleys in Romania where dragons roam, before settling down to plan their future together.
Ginny asking Harry out could lead to a very interesting chain of events. It could cause some jealousy among other students who have a crush on either of them. Ron, for example, might be a bit taken aback at first but then come around to support his sister and best friend. As for their date, it could be something unique like a midnight stroll in the Forbidden Forest (with proper precautions of course), where they talk about their fears, hopes and dreams, deepening their connection.
Ginny might be extremely embarrassed. She would probably blush deeply and try to hide behind Harry at first.
One possibility is that Ginny would be overjoyed. She has loved Harry for a long time. She might immediately say yes and then they would start planning their future together. They could think about where to live, perhaps near the Burrow or at Hogwarts if they both end up teaching there. Their friends would be very excited too, and there would be a big celebration at the Burrow with lots of Weasleys and their friends like Hermione and Ron all gathering to congratulate them.
In this fanfiction scenario, perhaps Dumbledore would witness the kiss. He might then take it as an opportunity to give Harry and Hermione some special task or mission, believing that their new - found closeness could be an advantage in the fight against Voldemort. This could lead them on an adventure where they have to rely on their new relationship and figure out their feelings along the way.
One possibility is that this kiss could be a spur - of - the - moment thing. Maybe they were caught up in the excitement of a victory in the Great Hall, like after a big Quidditch win or a successful defense against a dark arts attack. Then, they would have to deal with the consequences of their impulsive action, such as rumors spreading throughout Hogwarts.