Conan and Haibara essay.
For Conan's exact relationship, most of it happens in canon. For those who don't know, Haibara is a traitor for the Black Organization that shrunk Conan. She also made the poison that caused him to shrink, which she also took herself in a failed attempt of suicide. (Failed because she intended to die, but shrank, but hey, she saw a chance to get away from the Organization then and took it.) So they're both in the same situation, stuck as children and have a dangerous Organization that might try to kill them.
At first Conan is EXTREMELY wary of Haibara. For good reason. He doesn't trust her because she was a member and at first thought she was trying to trick them. Though he does learn the story of why Haibara left. Haibara had left because her sister was killed and the Organization never told her why. This added onto the fact Haibara never WANTED to make a poison to kill people. That wasn't what the drug was intended to be (though she doesn't say what it WAS supposed to be.) It's a bit later he learns that her sister was someone he FAILED to save during a case earlier on. He finally accepts Haibara did leave the Organization and wasn't trying to trick them when he felt her saw her real feelings; when she broke down after she saw him solve a difficult case and demanded to know why he failed her sister through her tears.
This is a big thing about Conan: he is FAR more forgiving a people than he'd lead people to believe. He will never forgive a murder because it's NEVER right to take a life. However, Haibara never intended to take a life and he can't fault her for not being able to stop the Organization from misusing her research. After he saw the break down, he saw not a threat, but a girl who had been in a difficult situation and had lost her last remaining relative and was scared. In this way, he was able to forgive her of any fault of hers behind the Apotoxin and move on from his negative feelings.
From there on, he decides to help her. He understands how she feels. He KNOWS what it's like to be scared the Black Organization will pop up out of any corner. By having to be someone you really are not. However, he also realized he knew two things she didn't. One was he knew what it was like to be in there situation and fear of recognition. The second was he knew she did not have a normal child life and he knew what one was. So he more or less took it as a side mission to get Haibara to actually LIVE. To always be cautious, of course, but he wanted her to be able to actually LIVE.
He knows Haibara has a horrible tendency of seeing herself as a liability and would throw herselves into the fire if she thought for a moment it could protect anyone. He DOES NOT want this to happen. Not because she is the only hope he really has for a cure, but because over the course of time he CARES about Haibara. While she is a constant reminder of his case and stress in life, she is his friend. She is the only one she can call a 'peer.' Oh, sure, some people know who he is, but they don't know what it's LIKE. She does and honestly, he's pretty sure just her presence alone has helped him keep working without losing hope.
Why? Because she is proof that the Black Organization is NOT some untouchable, perfect entity of darkness. She got away. She LIVED. Not only that, he is not only working for himself anymore. Since she is a reminder, he remembers he has someone he's working to protect. After all, more than once he's had an opportunity to get something on the Organization because they had gotten a hint about her and he used it to his advantage. (He's not intentionally using her as bait. He's just taking advantage of what would be a pretty bad situation otherwise.) And he always does everything he can to make sure she's not in danger during a trap. Though he REALLY hopes Haibara never gets the self sacrific idea in her head again. (Seriously, things almost get ruined because of it or it causes extra danger.) While Conan will always try to save any life he can, Haibara is one of the few lives he'd more than risk his life for just a small, insignificant chance of saving.
Like if Haibara had done as one criminal did in an early case and somehow threw him out of a burning building that the person decided as their grave, come hell or high water, he'd find a way to run in there a again and drag her out kicking and screaming. Even if the Black Organization figured out he knew where Sherry was, he knows he'd never tell where she is no matter what they did.
Basically, Haibara gets Ran level of protection. (Seriously, while he'd ultimately save Ran because it's RAN, if it was a choice between the two, there'd be some definite hesitation.)
Of Conan's confidants, NO ONE knows everything he does. If they all pulled together, they'd get it all. The one to know the most immediately is Agasa. Eventually Haibara is to learn everything as well. The only reason Conan really delays with her is because it either A) Will make her paranoid and try to talk him out of something he needs to do or B) invoke that self sacrifice tendency he REALLY hates.
By the point in canon he was taken from, he feels like he understand Haibara pretty well. While her secretiveness and paranoia about the Black Organization is annoying, he understands why and can certainly sympathize. Sometimes it's even nice because it keeps his head LEVEL and it's good to get warning and new information that's necessary to keep bad stuff from happening when she feels she has to give it. He also knows that, while Haibara might give him a hard time, she WILL help him. (Even if it costs a new designer outfit.) And really, she can be quite cute around animals.
Upon arriving at the Island, not much as so much changed as...shifted. (Though it should be mentioned his stubborn insistence not to believe in the multiple world thing was partly because Haibara did help him reaffirm that idea by agreeing with him. It's easier to believe when someone else believes with you.) On the Island, he determined pretty quick that the Black Organization certainly wasn't involved because they weren't dead and there was some way the people behind the Island could view the memories if others on the Island had an event that they could see each others. While it wasn't a TOTAL relaxation, it was a bit of one.
So much so for Haibara. His biggest concern for Haibara back home is to 'protect.' She is a main target and he doesn't want to slip up and let her be caught. However, with no Black Organization around, it's shifted to what else he wants to do for Haibara. Namely, encourage her to be a bit normal. He's been kind of waffling how to do this though since he can't just TELL her things. That'd more than likely drive her away from him and he doesn't want that.
The first 'real' step might have been after the High School event when Haibara expressed that it was nice to be a normal high school student. It was at this point he asked her if she realy did plan to take the cure when it was made. For him, the reason he wants to return most is obvious. Everything else in his life COULD be regained as Conan except for one thing: Ran. This is not the case for Haibara. Her family is dead and her life was basically ran by the Organization.
He made a suggestion that it would probably be more than all right if she didn't take it. He's not set on this idea or anything, but he thinks it would be all right for Haibara to relive her childhood the right away. He's had his once and he has Ran waiting. Though he did feel relief when she said she'd stick around once the Black Organization was taken down, whether or not she took the cure and planned to keep a connection with the kids. He feels it's a good sign. That she really is starting to feel actual attachments in her life which is something he thinks EVERYONE needs.
The next point was later, after Ran arrived. Haibara said she might move out to another room. Conan did realize why: Kazuha and Ran were reminding her of her sister. And Conan was strongly against the idea of her leaving their little....'family' for a lack of a better word because of that. He said she should stick around at least one because he thinks it'd do her a LOT of good. Mostly because it could help her deal with Akemi's death a bit better. Of course, this derailed into him sort of but not quite using her as bait.
He will be trying to help her come out of her shell a bit more in the future, and trying to avoid to be obvious about it, though he's going to make sure to always be there for Haibara when she needs him. Because Haibara is an important friend to him and is someone that, in the end, he trusts completely. (And trusts himself to know when it's a good or bad idea to tell her something.)
...also, like with Beato, SNARKFEST. Not quite the long windedness as he gets with Beato, but the two snipe at each other a lot. (Conan has a facetious love for his friends.)