Before going ahead, please be adivised that there may be unmarked spoilers in this section about the current events from the latest manga chapters. So please advance with caution.
Honey and those around him
Honey has a good relationship with pretty much everyone really, especially with everyone at the Host Club. Of course his closest bond is with his cousin Mori who is basically the one who understands him more than anyone.
In latest chapters, their parents remembered how Honey used to tease Mori into doing pretty much anything, but it's probably due to Mori's strong loyalty towards Honey because of their families' past. But Mori is pretty much always there for him, taking care of him and keeping him from making any mistakes. He is definitely Honey's most loyal and dearest friend. (well Mori and Usa-chan I would think)
As for everyone else, Honey gets a lot very well with each of them in a special way: he is kind to Tamaki while everyone else teases him (even though he sometimes does it too); with the twins, he and Mori have given them support when they fight or when they're trying to help one another plus they lend them an ear and understanding when they need it.
He gets along quite well with Haruhi and it's something sometimes the twins and Tamaki are jealous of when they're trying to get on her good side. As for Kyouya, they get along pretty much normally I think, they do share a couple of things in common but I think Kyouya is sometimes annoyed at how much the club spends on sweets because of Honey's overindulgence haha.
When it comes to his family, only his brother and father have been shown in both the anime and manga. His father seems to be an easy going guy, unfortunately there hasn't been a lot about their relationship except that Honey does respect him a lot and he is quite proud of him.
His younger brother Yasuchika on the other hand holds a grudge against him because of his decision to indulge in cute things and people often mentioning how much they look alike. Yasuchika even believed at one point Honey was being controlled by his plushie, Usa-chan, since it was completely inhuman to devour an whole 3 story cake in one night as a midnight snack.
Fortunately, Yasuchika and Honey have a short combat where Honey wins and gains his brother's respect. Even though it pained Honey that his brother had that grudge against him, he just wanted the very best for him, which is why he had decided to accept his challenge to fight him.