Jungle Rollers - Green
The Great Gate - Yellow
Upstream - Yellow
Rolling Stones - Blue
Native Fortress - Red
Road To Nowhere - Red
Boulder Dash - Purple
Cortex Power - Blue
Lights Out - Yellow
Jaws Of Darkness - Blue

Castle Machinery - Green (Optional - Extra Lives Only)

Green - The Lost City
Orange - Generator Room
Blue - Toxic Waste
Red - Slippery Climb
Purple - Lights Out
Yellow - The Lab

Blue - Hang Eight
Red - Snow Biz
Green - Ruination
Yellow - Sewer or Later
Purple - Spaced Out (Also requires all other coloured gems for path)

Blue - Turtle Woods
No box breaking
Red - Air Crash/Snow Go
Secret route of Snow Go by the secret portal in Air Crash
Green - Eel Deal
Secret route behind the Fake Wall near the group of Nitro boxes
Yellow - Plant Food
Beat the timer
Purple - Bee-Having
Brave to the top of a group of Fake Nitro Box stairs

Air Crash (w/ Alt Exit)
Road to Ruin
Diggin' It
Cold Hard Crash
Piston It Away (w/ Alt Exit)
Night Fight

A little extra info on how they work in Crash 2 following Crash 1's rules:

• Destroying all boxes in a level (dying is allowed). Three levels (Air Crash, Snow Go, Road to Ruin) have an alternate entrance through the Warp Room in the basement.
  
• In Level 3 a clear gem is obtained from beating the timer that triggers just before using the first surf board. You must make it to the end of the level before the timer stops to get the 2nd gem.
  
• Coloured Gem Paths. Not all, but some of them contain a clear gem. After collecting a coloured gem you can trigger a coloured platform in some levels. These platforms lead to new paths that contain boxes and in rare occasions a clear gem of their own.

Air Crash > Snow Go - The secret level exit is located at the first jet ski section. Instead of jumping on the jet ski, bounce on the floating crates to reach an unmarked stone platform on the right wall of the map. The Speed Shoes you unlock for defeating Cortex will make this task a lot easier.

Bear Down > Air Crash (Needed to get all crates) - At the very end of the level, after Crash is kicked off the running polar bear, turn around and use the floating circle platforms (that drop into the water) to reach a small island in the pond. Hop on and you’ll ride the secret teleporter.

Un-Bearable > Totally Bear (Secret Level) - Once again, this secret exit is only available at the end of the stage. Before leaving through the tunnel, hop across the hole and backtrack to find your tiny polar bear companion. Hop on him to teleport away.

Hangin’ Out > Totally Fly (Secret Level) - Dropping down through the second sewer hole, Crash will land in a pool of blue-hued water. Instead of continuing forward, move toward the camera (backward!) to reveal another hidden hole. Drop in and reach the end of one very tricky hanging sequence to reach the secret exit.

Diggin’ It > Road to Ruin - At the fork in the path, follow the right path and continue forward until you encounter a poison-spitting plant on a single island to the left of a narrow path. The platform is just big enough for the plant monster, so stomp it and you’ll immediately land on the teleport.

Red - Bone Yard
Purple - Tomb Time
Blue - Sphynxinator
Green - Gone Tomorrow
Yellow - Dino Might!

Red - Deep Trouble
Clear all crates then appears upon using the "!" crate you find at the end of the level.
Purple - High Time
Death Route
Blue - Tomb Wader
Death Route
Green - Flaming Passion
Death Route
Yellow - Hang 'em High via Secret Entrance
Secret entrance is in Warp Room 6 - requires that you have at least 10 relics obtained to access

Death Routes much be reached without dying. After completion and reaching a checkpoint, you can lose a life and retain everything you gained up to that point.

Eggipus Rex
Dino Might! - On the Yellow Gem path during the triceratops chase, let the second pterodactyl that appears carry the player away. The screen will fade and take you to the secret level.

Hot Coco
Road Crash - While racing keep your eyes on the left of the screen. When you see a yellow sign displaying an alien head, crash into it and you'll be teleported to the secret level.

Thanks IGN / USgamer / Powerpyx / Crash Bandicoot Wikis for the info.

Reworded, formatted and compiled into a quick and simple guide for ref while playing.
Makes some of the gem collection a little less ardious.

Have fun!