E1S1:
Go through the door, turn around and look up. There is a square window in the wall. The easiest way through it is on the left of the door. Just 3 jumps should get you into the opening. Now you can see the starting room beneath. Jump past it on the left or right. You'll end up on the rainbowroad on top of the map.

E1S2:
Go through the door, look up. The top of the pillar right infront of you is our goal. As with the previous one there are multiple ways to reach it. I've found the easiest one to be the one where you get on the arch on the right wall of the map, by jumping on the small broken pillar on the ramp and then shift+jump towards the arch. Get to the part of the arch closer to the exit and look towards the main pillar. You should see a rainbow ball on a platform. Shift+jump towards it and you should be able to reach the ball. Walk through it and you're on the rainbowroad on top of the map.

E1S3:
Go through the door, look to the left. The window above the pipes is our goal. Shoot the door open, go through and to the left. Jump on the three pipes and then up on the two pipes. Walk to the other side of the room, then jump ontop of the cube sticking out of the wall. Then ontop of the Wall with the door in it. Walk back to the other side of the wall, drop onto the two pipes and walk towards the window. Jump through it and out. Top of the map, rainbowroad.

E1S4:
Easy one. On the left side of the room, there are red cubes sticking out of the wall. Climb them up to the top. Yay rainbowroad.

E1S5:
Go through the door, turn left. It may be hard to see, but the black cube in the bottom corner of the wall with the door in it is sticking out enough for you to jump onto it and up again. turn around and jump onto the platform under the cubes coming through the roof. In the middle of the platform face towards the right side of the room and shift jump into a tunnel. Shoot the door open and run through it. You end up on top of the map and the rainbowroad. As a bonus you can go swimming with the crab and piranhas now.

E2S1:
Kill the worm go to the back and jump ontop of the stalagmite underneath the hole in the cave. You should be able to shift+jump through the hole and out of the map. Yay Rainbowroad.

E2S2:
Update: A better way was suggested: you can turn right as soon as you enter the room, hop up onto the platform and fall into the gap between it and the wall with the entrance. Drop down again and head towards the other side of the room. You'll find a rainbowball on the left halfway there. Jump into it to end up on the rainbowroad.

E2S3:
Through the door and climb the right glass ramp at the hight of the remaining number display. Face towards the left side of the room. You'll notice that the cieling is lowered towards you forming a ramp with black beams to each side which you can doublejump onto. Be weary that the glass doesn't have collision. Run along the black part towards the opening, double jump out and yay, rainbowroad.

E2S4:
Through the door and face left. It's another easy one. You can climb the green cubes up to the top of the left wall with occationally having to jump on a black one that just comes out of the wall enough for you to find a foothold. Once you reach the top left corner of the Room, look down towards the green cubes ontop of the door. See the rainbow ball, get it and enjoy another rainbowroad.

E2S5:
A lot of jumping on this one again. When you enter the room, you should notice that on the left side of the tree, there's a branch coming down far enough that you can jump on top of it. Walk along the branch to the trunk, then turn right, up there's the next one you can jump to. Repeat to reach the third level of branches. Face the left of the trunk and jump up to the fourth level. Now to get to the fifth easier you can find the branch that goes upward a bit, walk out on it and jump back to the middle. This works on the lower levels too. Now when you're at the top locate the wall that is pink, all others are blueish white. You want to go on the branch that faces away from it, follow it untill you hit the wall, and shift+jump over it. Yay rainbowroad.

E3S1:
In this room, there are 3 windows. You can reach every single one but the easiest one is the one right of the room (when facing the door). You can jump onto the black box, where the cubes disappear into, jump straight into the wall. Double jump to the right onto the next one, then turn around and double jump into the window. Jump out into the blue abyss to find yourself back on the rainbowroad.

E3S2:
The wall where the cubes come out from has a square, second from the bottom 9th from the left that you can jump on to to get on top of them. Now walk towards the left, along the left wall until you face the red block sticking out a bit. Doublejump around it. Looking at the wall with the door, you should see the two red cubes sticking out. You can reach both with a doublejump but the lower one is easier and you can jump up to the higher one from there. Now jump along the wall with the door towards the next red block. You can not exit the windows here, you have to continue along the wall to the red block right above the entrance. It might seem to far, but you can reach the red block in the corner of the room with a shift+jump easily. Now a tricky one, along the right wall you need to jump underneath the red block above towards the next one, be patient with your doublejump. Hitting your head means failure. Once you did it, turn around, jump on top of the red block above and this window you can walk through. Ronbowraad.

E3S3:
Closed room again and again you need to head to the left side to get started. On the lowest yellow cubes face to the back of the room and continue. Now you dont need to turn around, you need to jump on the yellow blocks sticking out of the backside of the room. Turn right and continue along that side. Now jump into the corner of the right and back side of the room onto the black cube sticking out a little. Turn and jump along the right side to the next yellow pair of cubes and in the same direction onto the next. Turn around and jump up, turn around and jump up again. You should see the rainbowball now ontop of the four yellow cubes on the wall with the entrance. Go get it and enjoy another ride on the rainbowroad.

E3S4:
Enter the doughnut and turn left. You notice the windows on the outer topside? Yea every single one of them is closed except the two on the right of the entrance (when facing the door). You only need to do one jump but it's a bit of a tricky one. Find the nearest highest spot in the center of the doughnut and shift+jump. Yay rainbowroad and you get to see the doughnut from the outside.

E3S5:
Whew, last one and an easy one too. To the right and the left of the entracne you should notice some dark poles sticking out forming a narrow stair. I'll leave the rest of the mystery to you.

Although not tied to an achivement, it's possible to break out of the starting area.
I climbed the wall next to the options pop up. And yes, there is a rainbowroad. :)

You now got all breakout achievements!
Firingmalaser,
Greetings,
M.Cheese