One of the unique things about Death Note is that its protagonist, Light Yagami, is also a villain; as a result, there is a mix of times when Light is right about things compared to when his opponent, L, is correct about things. Both these characters are geniuses, and the Kira case is one of the only things keeping them both engaged in life.
Even if L is the greatest detective in the world, there are still times he can be wrong or can misjudge a situation. Light, likewise, is only a university student, albeit an incredibly smart one. On the other hand, there are many times that the two of them are right, even if they can't do anything with their knowledge.
Light: Not Being Able To Pick Up The Notebook
After he loses his memories, Light demands of L whether he is the kind of person who would pick up the Death Note. The audience knows - and L suspects - that, yes, he is. However, without the memories of the notebook, and with the intellectual stimulation of working on the case, Light at this point isn't the kind of person who would pick it up. It's heavily implied that Light was depressed before he came up with his plan to rid the world of criminals, so having the Kira case to lean on with L alongside him is a massive change to his cynical side.
L: Light Is Kira
L suspects early on that Light is Kira - and he's right. He correctly deduces that Kira is a student and that he has a childish view of the world, both of which are true. When he finally narrows it down to Light, all he needs is proof - proof he never definitively receives. Regardless of that, the Death Note episodes are still gripping. The audience, and Light himself, know he's correct, but Light, therefore, does everything he can to stay just one step ahead of L in their endless game of cat and mouse.
Light: Naomi's Death
Light masterfully manipulates Naomi Misora into giving up her name so he can kill her, despite still just being a high school student. That element is fairly common for protagonists and makes the anime an easy one for newbies to start with. Light's right that she's managed to deduce who Kira is, and therefore he manages to stay ahead of L by killing Naomi. She's a clever former FBI agent - but he's just slightly smarter. No one else is aware enough of the situation to put a stop to it, and unfortunately, Naomi dies with neither her nor her fiance Raye avenged.
L: Kira Is A Student
Due to the hours of Light's early killings, it's easy for L to deduce that Kira is a student. None of the police officers seem to have taken any of those specific details (the hours when the murders were committed) into consideration at the meeting.
He turns out to be right since Light is still in high school at this point. It's an early sign of how L notices everything, and just how dedicated he is to the details that others might miss. Frustratingly, Light notices this too and starts changing the details of his murders.
Light: Takada's Death
Light manipulated Mello and Matt's kidnapping of Kiyomi Takada to his advantage - even though it meant her death. By that point, she wasn't useful as a pawn anymore. He was right that getting Takada to kill herself would also get Mello (and indirectly Matt) killed as well. Takada's death brings Light's plans closer to their conclusion, and therefore brings the series to the beginning of a close. Takada's death is nonetheless one of the darkest elements of Death Note.
L: Kira Is In Japan
L conducted his test with Lind L. Tailor to get Kira wound up, and it worked. He was also testing where Light was since the message was only broadcast in the Kanto region. Both of these tactics worked, and L knew where Light was from that moment on. L, therefore, learned from these sessions just how to push Light's buttons, as well as valuable information about where, precisely, to look for Kira.
Light: Mikami's Methodical Nature
Light - though he considered it at the wrong time - was correct about Mikami's methodical nature. When Mikami changed his routine because he thought Light would require it, that, in turn, changed Near's plan. This allowed Near to prove that Light was Kira to his face. Light was perfectly correct about Mikami - and it's part of why he was chosen as X-Kira - he just chose to think of the information at the wrong time. Though he might be one of the smartest characters in the series, Light still has the bad luck to be underestimated.
L: Misa Is The Second Kira
L arrests Misa on suspicions of being the Second Kira - and though she erases her memories of her time as Kira - he is correct. It seems a strange choice since she's an airheaded model, but her closeness to Light helps prove it to L, though he has no actual proof.
He does try to get it out of her, including by keeping her in a straightjacket for days. However, since she really doesn't know anything, L can't do anything about his correct suspicions.
Light: Misa's Lovelorn Nature
Light quickly realizes - correctly - that Misa is so enamored with him that she really is willing to do just about anything for him. Kira killed the criminal who murdered her family, after all. He, therefore, strings her along for the rest of the series, dangling his love in front of her like a carrot in front of a rabbit. She is his greatest asset - and one he uses for years. In the process, though Light doesn't care, Misa is ruined as a person.
L: Luring Higuchi Into The Open
L makes the right moves to lure the Third Kira, Higuchi, out into the open. With Light, Matsuda, and Wedy's help, he is able to narrow the Yotsuba Group down. After they've determined that Higuchi is the Third Kira, he uses Matsuda to make sure Higuchi is suspicious enough to try to kill him. Eventually, he is caught, and L confiscates the notebook. Though he gets help, L ultimately plans everything that leads to Higuchi's arrest. WIth Light's altered state due to having lost his memories, he and L are on the same side. However, it's clear that L is the one in charge of the investigation. Though L makes the right decisions to ultimately apprehend Higuchi. unfortunately, the plan ends as Light intended: with his memories of the Death Note returned.