The Sonic series does a good job with this, and will point two two characters:
Knuckles:
He is introduced in Sonic 3 with the following:
He KNOCKS OUT SUPER SONIC like nothing XD! He takes the emeralds and chuckles :').
Not to mention he was always in your way at every turn in Sonic 3. But once you fight him one-on-on in Hidden Palace Zone, he learns that Robotnik really is the bad guy and gets badly hurt trying to get the Master Emerald back. So, he lets Sonic move on to Sky Sanctuary Zone.
From this point onwards, he grows to respect Sonic a bit more and while they still but heads sometimes, he really is a great buddy to Sonic.
The recent Boom Cartoon has a lot of moments where he and Knuckles really bounce off each other perfectly XD!
Boom does characterization well (despite making Knuckles dumb :l).
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Blaze:
She is introduced as a new rival to Sonic in the Rush series and in many ways, is the complete contrast to Sonic: She uses fire attacks (vs. Sonic using the wind), she is a guardian and more reserved (while Sonic is more open and free), and she is very hostile toward everyone (while Sonic is calm until the situation calls for action.
Sonic's interactions with her were great in the original Rush and their fight in that game was great:
Her character in Rush Adventure was enjoyable too and her interactions with Sonic & Tails were quite nice .
Marie can go in a hole X(. So annoying
Shame Blaze got few major roles in the series post Rush Adventure (Black Knight she had a role and she cameo's in Generations HD and Colors DS).
Kinda like Knuckles, she was not used as much in recent games.
Jowy, best friend and destined rival the day the both of you took on the rune of beginning. He contributes to a major part of the story and plays many major roles that directly affect the results of the war.
In terms of gameplay, I think Chain of Memories did a good job at presenting him as an obstacle to Sora. With the way progression worked in that game, the player can learn to anticipate a boss fight, and when it's Riku, there is a good amount of tension.
In the story he serves as a foil to Sora, with how he walks the path of darkness in contrast to Sora's light. They have the same goals in the first game, but their circumstances differ, with Sora meeting Donald and Goofy and Riku meeting Maleficent. Then Riku teams up with King Mickey, and seems to always be working with a clear goal in the shadows, while Sora tends to get swept up in the main plot. Dream Drop Distance is the only time where it seems like they're on the same path.