Knuckles doesn't converse with a ton of characters (He's a loner, remember?), but has had increased interaction with the two below.
Doctor Robotnik (or Eggman) – Knuckles finds himself fooled again and again by Doctor Robotnik, as Doctor Robotnik tries to steal the Master Emerald in Sonic 3, and then again in Sonic Adventure. It's no surprise that Knuckles, like any of the good guys, can't stand Robotnik, but he trusts Robotnik when he says the Emerald is in danger. I don't think it's because Robotnik is trustworthy in any way; Knuckles is just gullible, and won't take chances when it comes to the Emerald. The two don't interact much past that, and even though they were on the same team in Sonic 3, they don't like one another; Robotnik was just using Knuckles. Knuckles has joined Sonic's plight to defeat Robotnik many times.
Tails - Knuckles and Tails have a secondary relationship at best. They're on very good terms, though, and work very well together when teamed up with Sonic. Knuckles has acknowledged and appreciated Tails' brilliance and tech-savvy nature on multiple occasions, and Tails has noticed Knuckles' strength. I think the two could become good friends if given the opportunity, but they are always helpoing (or getting in the way of, in Knuckles' case) Sonic, and when not saving the world, Knuckles is committed to living alone and guarding the Master Emerald.