I know this topic is probably no longer in effect, but I still wanted to try an put in my two cents worth in as I just recently got into the Sonic groupings. And I am aslo intrigued in that everytime Sonic Team introduces a new character they always seem to be in groups of three, each representing the Speed/Flight/Power combination, so here are my suggestions. I started with the four in Sonic Heroes and added trios already formed in other games, and made some of my own, from pre-existing characters.
From Sonic Heroes
Team Sonic - Sonic the Hedgehog, Miles' Tails' Prower & Knuckles the Echinda
Team Rose - Amy Rose, Cream the Rabbit & Big the Cat
Team Dark - Shadow the Hedgehog, Rouge the Bat & E-123 Omega
Team Chaotix - Espio the Chameleon, Charmy Bee & Vector the Crocodile
Teams that have already worked together (I'm creating some of these team names)
Team Metal - Metal Sonic, Tails Doll & Metal Knuckles (In a game their story would be similar to whatever Team Sonic's was but with opposing goals)
Team Fang - Fang the Sniper, Bean the Dynamite & Bark the Polarbear (Their levels would be puzzles, but with monitary goals)
Team Babylon - Jet the Hawk, Wave the Swallow & Storm the Albatross (Their levels would be aerial only)
Teams that could work together
Team Blaze - Blaze the Cat, Silver the Hedgehog & Marine the Raccoon (I do not know if Silver and Marine have worked together before or not, but I know Blaze has worked with both Silver and Marine at different times, which is why I put her as the leader of the group) (Their levels would involve traveling through time and space)
Team Gamma - Bomb, E-102 Gamma & Heavy (Three Robots that have rebelled against Dr. Robotnik) (Their levels would be like E-102 Gamma's in Sonic Adventure, freeing other Badniks, just this time Gamma will have help)
Teams that I just threw together
Team Little - Classic Sonic, Yacker & Chip (All introduced around the same time) (As they are all little, their levels would involve them going into smaller areas that other teams wouldn't fit)
Team ??? - Mecha Sonic, Honey the Cat & Chaos (These characters have only appeared once in games, Mecha & Honey were both in Sonic the Fighters, Mecha as a hacked character & Honey as a latter addition, Chaos was the villain in Sonic Adventure, but he is now neutral) (Each character were single characters that I put together because, being together, they fit the Speed/Flight/Power combination.
Team Echinda - Shade the Echinda, Wechnia the Echinda & Tikal the Echinda (all echindas from different eras of Sonic's career, Wechnia is actually a glitch character from Kunckle's Chaotix)
Team Ristar - Mighty the Armadillo, Ray the Flying Squirrel & Ristar the Shooting Star (Mighty & Ray are from the Sonic Franchise, Ristar is from another game. Both Mighty and Ristar were born from the same idea that created Sonic, and Ray was born from the same idea that created Tails). (Their levels would involve Mighty & Ray finding themselves in Ristar's world [a possible way of explaining why they haven't been seen in any recent Sonic game], and Ristar deciding to help them find their way home.
Team Flicky - Tiger the Tabby Cat, Flicky the Bluebird & Iggy the Iguana (Before Flicky became a Badnik, she starred in her own video game with Tiger & Iggy being the main antagonists; however, since they only three characters in the game with Tiger, Flicky & Iggy, I decided to put them together in the game)
Team Outsiders - Ulala, Amigo, NiGHTS, Aiai & Billy Hatcher (a five person team, although only three would be seen at a time, all are obviously from different games; however they have appeared in Sonic games, with NiGHTS having the most appearences, which is why NiGHTS is also the leader)(The preferred combinations would be either NiGHTS, Ulala & Aiai [who appeared together in Sonic Riders], NiGHTS, Amigo & Billy Hatcher [who appeared together in Sonic Riders 2], NiGHTS, Amigo & Aiai [NiGHTS and two monkeys] or NiGHTS, Ulala & Billy Hatcher [NiGHTS and two humans] (Their levels would involve them teaming up because their worlds are merging due to cause and effect caused by Dr. Robotnik & Dr. Wily)
Team X - Megaman X, Axl & Zero (The upcoming Sonic/Megaman crossover is the main reason I started creating these groups)
Team Y - Alia, Pallette & Layer (I couldn't think of a better name for the female counterpart to Team X)
Team Jackrabbit - Jazz Jackrabbit, Spaz Jackrabbit & Lori Jackrabbit (Just wishing, I have enjoyed the Jazz Jackrabbit series, and as I recognize Jazz being a copy of Sonic [just with guns] I would love to see a Sonic/Jazz crossover.
Bonus Teams (These characters, all from the Sonic Games would come together to form special teams. Maybe for like bonus levels when the game was done.
Team Hedgehog - Sonic the Hedgehog, Silver the Hedgehog & Shadow the Hedgehog (this could be the team that works together to defeat the final boss)
Team Cat - Blaze the Cat, Honey the Cat & Big the Cat (this is more of the bonus team that would be playable after the game was over)
I tend to put a lot into my ideas, sometimes I have to work them over and over until they are more smoothed out. I do enjoy criticism, so please criticize.