There are 70 coins, 36 of which come from the 36 main levels from 1-1 to 6-6.
You can see which of these you have by pressing tab in any hub area.
This guide is a list of every other coin, in order of when you can first acquire them.

[NOTE: For unlocking this achievement, I would recommend getting an in-level coin last.
The first time I tried, my 70th coin was in a secret tunnel in a wall, and it didn't unlock.]

Sanctuary
(1) at the tombstone that tells you about shards on the left of the level, fall or ledge-dash left to find it at the bottom of a ravine.

First nightmare
(2) get the gem and backtrack to the beginning (or finish and restart).
There is an upwards tunnel and some spiky platforms to jump across, and a key. Shine makes this easier but isn't necessary.
The key unlocks a door at the bottom of that tunnel from before, which leads to the coin.

Area between Sanctuary and Treetops
(3-7) 5-coin level.

(8) underneath the large platform with three smaller platforms, accessed by ledge-dashing and mid-air jumping.

Treetops
(9) between 2-5 and 2-6 at the top of a tree.

Second nightmare
(10) finish and reenter. At the beginning to the left, go up and shine through, and follow the path to it. The key door lets you finish the level, or exit through the new entrance.

Treetops
(11) above 2-2, shine the monuments in the order 2-3-1-2-3-1 to unlock. (Hint said by small creature in level.)

Sanctuary
(12) shine the monuments (not the gem holders) on top of the level in the order 1-1-2-3-2-3-1-3 to unlock. (Hint at bottom-right of level.)

Area between Sanctuary and Cavern
(13-17) 5-coin level.

(18) secret tunnel near top-left. Two openings, one above the tombstone near the line of boosters, one to the left of the big room after the single booster.

(19) secret tunnel near bottom-right, next to right-most tar pit.

Cavern
(20) above-right of 3-4, there is a secret tunnel indicated by patterns in the ceiling.

(21) near 3-6 there are five monuments. Shine them top-left, mid-left, mid-right, top-right, bottom, to unlock. Appears to the bottom-right. The hint is in the very bottom-right of the area:



Third nightmare
(22) to the right of the exit, across a gap.

Area between Sanctuary, Frozen Peak and the Temple of Techno
(23-27) 5-coin level.

(28) jump from the middle island to the far-right wall, there is a secret tunnel.

Frozen Peak
(29) to the upper-right of 4-1, two monuments. Shine 1-2-1-2-1-1-2-2 to unlock. (Hint is in 4-4.)

(30) above 4-4, there's a ledge, and above that is another ledge with a secret tunnel.

Fourth nightmare
(31) above the gem.

Temple of Techno
(32) to the left of 5-4, jump further left to a small ledge.

Fifth Nightmare
(33) exiting the first tunnel, head to the bottom-left through a narrow corridor and across a pit.

The Dark Sky
(34) ledge-dash right from the top of the elevator to a small platform under the main island.