Vroom!: At the very start of the Prologue (The Edge), you will land on a building on top of another building. Jump down onto the bottom level. The roof you are standing on now can be circled all the way around via a narrow ledge at the back. Be wary though, as you cannot lose momentum at all. So this means no sharp turns, no jumping and especially no crashing into things. You can also obtain Vroom! in the Jacknife chapter, in the canals. Just remember, do not stop moving.
Up, Over, Under, Onwards: Again at the start of The Edge, you will be facing a set of solar fields. Jump down onto these and use one of the solar panels to propell yourself over the fence, which you should coil over. Then, there is a silver pipe on the other side which you can slide under, granting you the achievement/trophy.
Free Flowing: In the training area, follow Celeste onto the balance beam. After you've crossed this, look to your right, where you should see a wall to wallrun off. Run onto it, then wallrun, turn, jump and speedvault onto the naighbouring building. This is also accessable in the Playground 1 and Playground 3 Time Trials, and is in fact a useful shortcut to keep your momentum.
May I Have This Dance?: See the Speedrunning page for more info, under the New Eden or The Edge sections.
Aaaand Safe! - In the Playground 2 Time Trial, after you jump off the planks and land on the orange catwalk, you will see a set of planks propped onto a box on a building to your left. Use this to wallrun and jump over the fence. (Remember to coil, otherwise A: You won't make it and B: You'll miss the Achievement/Trophy) After this, skillroll at the bottom and the achievement is yours.
Head Over Heels, Untouchable and Pacifist: The Edge counts as a chapter for these achievements, and since it's shorter, you should use this to your advantage. Remember for Head Over Heels to skillroll upon exiting the vent. (It helps also to face the direction you will be running in, so you roll out that way) Also, for Untouchable, you will encounter two blues shooting at you with pistols on a rooftop. Turn to the left, and there is a small narrow gap between the building and the edge, which you can run behind without the risk of getting shot.
Martial Artist: The easiest way to get this achievement is, again, in the prologue, after Celeste's cutscene. Four PK guards will be shooting at you with assault rifles, and you should run towards them, wallrun off the building next to them, jump off and kick them. You can even exit the scene of combat without dying.
Sweet Goodbye: In Jacknife, after you exit the drains, four guards will start shooting at you. you should run back with a reasonable run-up and jump back into the drains. Then, turn 180 degrees. If you start falling backwards and Faith flips off to the cops, you've got the achievement.
Hey, It's-A-Me: In The Edge, get onto the building with two blues shooting at you. To the right of the first guard, there is an electricity box, which you should jump on. Then, face the guard and jump. Once you are directly over him, press attack, and you should earn the Achievement/Trophy. This also works well in Kate, when you are running down the stairs. Just jump off the stairs onto them. As long as you are attacking the blues from a higher point than them, you should be fine.