One thing I would do, gameplay-wise, is make more use of parkour, and emphasize how Sonic moves through the environment, in a way that feels like the opening and ending of Sonic CD, and the opening of Unleashed, and so on. And other characters would follow that same mentality.
But mostly, I'd focus on the plot, story, continuity, characters, and lore. So in regards to that....
Tails' Adventures and Knuckles' Chaotix would be made 100% indisputably canon, they already tie in perfectly to the 1-CD-2-3K plotline as apart of Tails and Knuckles' stories. I'd also include Tails' SkyPatrol, Sonic Triple Trouble, pretty much every game. The story of the series as a whole would feel interconnected and ongoing between each game, several adventures tied together into one big adventure. From the start, I'd cement Amy's spot as a 'main character' among the group, even if Sonic and the others don't immediately treat her as such. I'd develop her talents in mysticality and expand on that side of her heavily, in a way that is somewhat similar to how Nimue was portrayed in Sonic and the Black Knight, in terms of role in the story and stuff. Never Lake, the Time Stones, and Little Planet would be important to her character to some degree, and would not just be some random setting and stuff for a single adventure and forgotten about afterwards. I'd find a way to link Amy and her fortune telling to the Time Stones' power over time, and thus fate and destiny, and I'd possibly find a way to symbolically link Amy Rose to the Miracle Flowers that sprout from the seeds that come out of the badniks in the game.
For Knuckles, I'd have him as a character that is present in the story from the very beginning, in Sonic 1, but not fully explained or revealed what's going on or who he is. Basically, I'd have Knuckles appear at the very beginning before the actual story starts, in the Hidden Palace, sensing something odd with the Emeralds or something before they disappear, or if they are already gone, just sensing something among their power in general, and I'd have Knuckles have his own story in the background from then onwards, occuring on Angel Island and stuff simultaneously to Sonic's battles with Eggman up until the plot of Sonic 3 and Knuckles. I'd do this as a way to really try and establish Knuckles' role as Sonic's rival, the mountain to his wind and all that, the character that clashes with Sonic as an enemy before eventually joining the group as his best friend and stuff. But from 3K-onward, I'd make sure it's clear that Knuckles still doesn't entirely trust Sonic until between Sonic Adventure 2 and Sonic Heroes, and I'd delve in to the supposed "jealousy" he has of Sonic's freedom, and how he'd percieve someone like Sonic, a guy who's a reckless daredevil who always does whatever the heck he wants. But above all else, I'd make it very clear that Knuckles is not an idiot. I'd show that Knuckles is clearly a smart individual with a lot of sacred, ancient knowledge, as well as being able to make use of all the ancient advanced technology his ancestors left behind. It has also recently come to my attention that someone with Knuckles' experiences and life could NOT grow to be a gullible person, but the exact opposite, someone who's extremely distrustful. But at the same time, I'm aware that Knuckles is meant to be naive. Knuckles' rivalry with and perception of Sonic will tie in to Sonic pretty much being an embodiment of chaos, and how dangerous someone like Sonic is. I'd also have Knuckles start out putting his duties to the Master Emerald above all else, to the point that nothing else would really appear to matter to him that much. That way, Knuckles would have to work his way up to being the heroic and powerful echidna we know him as today. I'm not sure if I'd have Sonic Chronicles as it is be canon, or if I'd just heavily alter the story severely and have it focus more on Knuckles, as a Knuckles game perhaps, since the story focuses heavily on the Echidnas.
For Tails, I'd make sure that the character arc from Sonic Adventure continues onwards as Tails learns to be more inependent and believe in himself more, growing as his own hero by learning from everything Sonic ever taught him. So that by the end of Sonc Heroes, he'd be fully capable of being a hero in his own right and standing alongside Sonic as an equal and not as a simple sidekick. From there, Tails would often have adventures and stories on his own without Sonic, or playing a much bigger role in Sonic's adventures then before, and not simply being support for Sonic. But at the end of the day, supporting Sonic is something Tails would never stop doing as a character. I'd also play around with the idea of Tails being a kitsune.
In terms of the universe as a whole, I'd draw from every official continuity and merge elements from each of them across Sonic media into one single cohesive universe but with the games' canon as the dominant one, similar to the comics right now. But in regards to Sonic himself, in terms of identity and family, I'd mostly draw from the manga Amy and Charmy came from and have Nicky Parlouzer be Sonic's real name, and stuff. But I'd also play around with the idea that the identity of "Sonic" is or was some entity of it's own that was originally separate from Nicky Parlouzer's original personality, and I'd try to tie it in to that story about a guy that died while breaking the sound barrier in a supersonic plane and came back later as a blue supersonic hedgehog like the emblem on his flight jacket. Like, A test pilot nicknamed "Hedgehog" dies in an accident while breaking the sound barrier in an aircraft, his soul takes the form of a blue hedgehog in the wind or something, and many years later, the blue hedgehog soul thing bonds with Nicky Parlouzer, or it's a reincarnation thing or whatever, and at first it's like 2 people in one body but eventually they become one and the result is the personality of Sonic we all know. Or something like that...
Sonic Adventure's subplot between Sonic and Shadow, about Sonic being the true Ultimate Lifeform, would be expanded upon and carried on through each game afterwards, building on it and explaining just what Sonic really is and how. Shadow's game would be a little more linear, but it'd still have multiple endings based on choices. It's just that there'd be a path that is OBVIOUSLY the true canon path, but the alternate pahs wouldn't be too different other than Shadow's choices and his conclusion at the end. I'd also expand on the subplot between Shadow and the Commander, and use that to build up to Shadow joining G.U.N in the future more obviously than vague training in Expert Mode. Afterwards, Shadow being an agent of G.U.N and working for the military and president would become intertwined with Sonic and Shadow's rivalry, and becoming a representation of one of the main differences between them, and would definitely become a reason that they would clash and fight even if they are both technically trying to do what they think is right.
The plots of Sonic '06, Sonic Rush, and Sonic Rivals would all be connected and linked together in such a way that the storylines of Sonic Rush and Sonic Rivals would clearly feel like the results of the timeline change in Sonic '06 and stuff, and Eggman Nega's ambitions would be to recreate the corrupted Solaris. Silver essentially being another Sonic would be made more clear and expanded upon. I'd firmly establish that Silver is Sonic's future reincarnation, or something along those lines, and thus he shares Sonic's speed and abilities, in addition to Psychokinesis and ESP. But in regards to Speed, Sonic CD's timetravel mechanic would become apart of Silver's abilities and even gameplay. Silver would time travel by reaching high enough speeds to break the time barrier. And travelling to the past to change the future would be apart of Silver's gameplay as well. The Sol Dimension's existence would be explained (eventually) as a result of Sonic '06. The circumstances of Solaris' erasure resulted in the energies of Solaris, and Blaze and the Sol Emeralds, entering another dimension and transforming it into a world linked to Sonic's like the other side of a magnet. Eggman Nega being so interested in the Sol Emeralds and Blaze's world despite being from Silver's time would be because of this. He'd need the Sol Emeralds and the Jeweled Scepter to recreate the corrupted Solaris. Also, Marine having water powers would be a non-vague thing.
One aspect of how I'd structure the... "eras" and flow of events, and storyline/plot structure and all that, would be based around the four elements, going from Wind, Water, Flame, Gaia, and back to Wind. The Water theme would start with Sonic Adventure and Chaos (and continue through the rest of the Dreamcast Era, especially in regards to Chao, the Artificial Chaos, and stuff like that), the Flame theme would start with Sonic '06 and Solaris (and continue through Rivals. Rush, and Secret Rings), and the Gaia theme would start with Sonic Unleashed and Light/Dark Gaia (not exactly sure how that'd be expanded, though...), and the second Wind theme would start with Sonic Riders and the Babylon Rogues, and also include Black Knight. The first Wind theme would mainly be based around Sonic himself, as he's like the wind, during Sonic 1-3K and so on. I'd also like to add that Sonic would continue to wear Chip's necklace on his wrist. I'd probably also have a way to give Sonic an elemental-based upgrade souvenir thing each time, with Chip's necklace acting as a source of Gaia energy Sonic could use for something earth-based, somehow.
If Project Sonic 2017 is what I think it is, then I'd merge it with Sonic Generations somehow, at least as 2 adventures that flow directly into each other, if not the exact same adventure.
Eh, I guess I'll end this here. I'm not sure what else to add...