The story of Sonic Adventure 2: Battle is quite long and multiplex. This is due in large part to the fact that it is split into three seperate perspectives: that of the heroes, that of the villains, and, finally, that of both forces combined. Below I will give you an in-depth recap of the entirety of the story of the game completely from memory. Here goes:

Hero Side Story

The hero side story starts with Sonic (the hedgehog) escaping from captivity inside a military helicopter. How he was captured in the first place is never detailed and, frankly, is outside the scope of this essay. Sonic proceeds to plummet down to the city below - which I'm sure has a name, but can't for the life of me recall ever hearing.

After hitting the streets on a conveniently sharp slope, Sonic skates away from the police on debris from the helicopter, but still gets caught by a military war mech, because the government has decided it really doesn't like him. Shortly after handily defeating the sheer power of weaponized taxpayer money, Sonic finds Shadow the Hedgehog standing atop a nearby building. Shadow shows Sonic that Chaos Control exists, which is the ability to harness the power of a Chaos Emerald to teleport, or go really really fast. The game isn't exactly clear on that in this scene.

Sonic ends up getting arrested by the police, and Tails ends up breaking through legions of the US's finest to free him, although he gets distracted by Eggman showing up and Amy ends up being the one to break Sonic out. At some point before this, Knuckles smashes the Master Emerald to keep Eggman from stealing it.

After Sonic and Shadow battle on the prison island and they both realize Eggman has rigged the island to blow, they race to get off the island. Sonic ends up hitching a ride with Tails (I think) and they escape at the last second. Fast forward a bit and Eggman starts broadcasting messages to the world about how he wants to take it over. Sonic and tails have a meeting about what to do and decide to break into the president's limousine while it is driving down the highway. They come to this decision right after Knuckles climbs out of a manhole in front of them, having been at some point before this searching for Master Emerald fragments in the sewer.

Once they rob the president's limo, Sonic and Co. go to the desert where they tracked Eggman's call signal, or something. They break into his secret pyramid base and find his rocketship, stealthily riding it into space. At some point before this, Knuckles searches for keys to get them through a door.

Once in space, they are taken to the Space Colony ARK, which houses many things, including: 1) a giant biological horror, 2) a planetary laser capable of destroying the earth, and 3) painful memories of Shadow's traumatic past. Once here, a plan is hatched and set into motion to sabotage the laser with a fake Chaos Emerald. Meanwhile Knuckles searches a nearby meteor for pieces of the Master Emerald, having at some point before this spilled the pieces he had into space.

While Sonic and Tails try to deploy their foolproof plan, the are caught by Eggman, who traps Sonic in a glass tube and uses trickery to get Tails to admit to having a fake Chaos Emerald. He then ejects Sonic in a pod set to self-destruct, so Tails gets upset and fights Eggman. While these events occur, Knuckles finally boards the ARK, having at some point before this fought Rouge in space for a few pieces of the Master Emerald.

Dark Story

I don't remember much about this one. Just kinda invert the story above.

? Story

Unbeknowst to the others, Sonic manages to use Chaos Control to escape the pod Eggman ejected him in and appears in a corridor of the ARK. For some reason, everyone decides to stop fighting so they can disable the cannons - even Eggman. I'm pretty sure it's related to an evil plan set in place by Eggman's grandfather. They realize that the only way to stop the cannons at this point is to infiltrate to its very core, and hatch a new, better, more difficult plan. At some point before this Knuckles sees Sonic appear in a corridor in front of him.

One thing leads to another and there's a massive mutant lizard thing that must be destroyed. Amy, along with memories of Maria, convinces Shadow to save humanity instead of destroy it. Sonic and Shadow team up as Super Sonic and Super Shadow to take down the Biolizard in space. In the end the team is triumphant and everyone is happy, although Shadow dies. Or does he? At some point during all of this, Knuckles comes to the decision to abandon his sworn duty at Angel Island so he can participate in Sonic Heroes.