Sonic Adventure DX Director's Cut

Sonic:
Light Speed Shoes - Sewers of Station Square
Crystal Ring - Hotel in Station Square
Ancient Light - Mystic Ruins near entrance to Red Mountain
Tails:
Jet Anklet - Near entrance to Twinkle Park
Rythem Badge - Mystic Ruins (past)
Knuckles:
Shovel Claw - Near entrance to Mystic Ruins Chao Garden
Fighting Gloves - Jungle area of M.R.
Amy:
Warrior Feather - Win Hedgehog Hammer mini-game in Adventure Mode
Long Hammer - Same as above but it must be after the Egg Carrier crashes but before facing Zero
Big:
Power Rod - Under Big's bed in the Jungle
Life Belt - En route to Ice Cap stage
Lure Upgrades - Station Square, Mystic Ruins Jungle, Ice Cap Stage, and Egg Carrier
Gamma:
Jet Booster - Ammunition room (Egg Carrier)
Laser Blaster - Water Tank (Same adventure field but Egg Carrier must be at sea, not air)

Hope this helps.

Well, it depends on what characters you're talking about. I've only beat the game with Sonic, Knuckles, and Amy.
But, here are they're power-ups.
Sonic:
Light Speed Shoes: Station Square Sewer
Crystal Ring: Station Square Hotel (you have to light speed dash)
Ancient Light: Entrance to Red Mountain
Knuckles:
Shovel Claw: Close to M.R Chao Garden
Fighting Gloves: On a cliff in jungle (you need to climb)
Amy:
Warrior Feather: Beat the high score on Hedgehog Hammer
Long Hammer:Beat your previous high score on Hedgehog Hammer