Link to the Song:
   • Sonic and Tails R  All Around (Episo...  
I'll only be going over the essential parts of an excellent speedrun.
Emerald Route:
White, Cyan, Red, Green, Yellow, Purple, and Blue.
White:
From spawn, go to the tree shown in this time stamp 0:10, and climb it until you're sure you can jump to the elevated platform next to it.
Cyan:
The rail at 0:32 is difficult to skip, but possible. The moment you hit the very top of the rail, jump off and try to land on the cliff shown in the video. Continue on with the stage normally until you reach the spring just outside of the cave. Use the spring's power to climb over the mountain. From there land on the platform with the rail and continue normally. You may also skip the final bottomless pit by combining boost and jumpdash.
Traveling to Red:
Go through the tunnel and use the rails because rails make you faster.
Red:
At 1:33 do a jumpdash onto the left wall and keep climbing until you reached the top and jump towards the rail and skip basically 75% of red obby. I made a mistake at 1:41, but this route works, you can skip most of the homing attack targets. At 1:44 make sure to do a jumpdash before you reach the end of the rail to save a bit of time climbing over the ramp.
Green:
Use the volcano to get some height and jump on the trees like shown in the video 1:59. You need to jump from tree to tree to reach the spring and grab the green emerald.
Yellow & Purple:
I have no tricks here, just align yourself with the emeralds and try to boost the moment you get out of the teleporting phase before and after yellow.
Blue:
From purple, use the volcano's ramp to easily go over the wall that surrounds the city and make your way to the park. The tricky part here is doing timing your jumpdash over the wall that surrounds the park.