Is there any reason why Lelouch didn't go to his father to find out the truth about his mother's death as soon as he got Geass?
You can't just walk into the throne room of the Britannian emperor. Bear in mind that Lelouch was no prince anymore in title, he was banished (or something like that, I can't recall the exact term). It's like walking in the room of Queen Elizabeth or the POTUS; it's not something you can physically do. Even with Geass, he'd just be shot down by a sniper or Knightmare.
In Code Geass: Lelouch of the Rebellion Lelouch wanted two things: To find out the truth about his mother's murder and the complete destruction of Brittania. Lelouch could have stormed to the throne and tried to take care of his first ambition right away however he knew that would not solve his second problem so he decided to take the long route which was to overthrow Brittanian rule in the areas by taking over a terrorist group and branding them as freedom fighters to gain popularity and support in his quest to disband Brittania and create anew and although it took some time he successfully got what he wanted.