How do you achieve the textures on the right side of the mountain, where the noise seems to leave the shape. Is there a gradient applied, because there's slight differences in colors in places. And on the moon, how do you make it look like rough brush strokes?
Illustrator gets easily overloaded if you have something this complex as vectors. Every grain must be an object. You may create them with sprayers, scatter brushes and by tracing raster images which contain noise, but finally you still have 100000 objects instead of a few megapixel bitmap image.Try at first to make the textures in Photoshop. If you have resolution high enough you can use them in Illustrator as raster images without making them vectors.
Modern Illustrator versions even allow you to use bitmap images as brushes, so you can draw with in Photoshop made or scanned real brush, pen or crayon strokes.
Let's try to make in Photoshop something like the moon.
We start with dark grey background and fill a circular selection with color to a new layer. The circle diameter is about 350 pixels:
Blending mode Dissolve makes randomly one pixel holes to the layer. The hole density increases if the layer opacity is decreased:
If we insert a layer mask we can easily control the opacity without deleting our original shape. An easy way to get the mask is to reselect the circle and click "insert layer mask" icon in the layers panel.
Here the layer mask is inserted, the layer opacity is returned to 100% but the mask has got a few px gaussian blur. It makes the black to white transition fuzzy and the edge gets a hole gradient:
If we paint manually with low opacity (a few %) soft brush some black to the white area in the mask we get the wanted roughness to the whole fill. In the right there's also shown the layer mask:
You can adjust the effect depth by applying Curves to the layer mask. To have bigger holes you should use lower resolution or paint full black dots to the mask with a scatter brush. The holes generated by the blending mode dissolve are 1x1 px.
The mountaintop can be made basically as a black shape on a dark grey background. A part of the black shape is partially covered by a pink shape. These all are in separate layers which are fully opaque and have blending mode =Normal.
The shapes are made by filling polygonal lasso selections with the paint bucket. The paint bucket had tool blending mode=dissolve instead of normal. It made the edges grainy due the transparency in the edge anti-aliasing.
Grainy textures are added one by one. Here they are all visible:
The black shape has got a little noise with Filter > Noise > Add noise.
The pink shape has got three different manual paintings to layer ON_PINK. They all are made manually with soft brushes which had tool blending mode = Dissolve. The colors were black, orange and red. A selection prevented painting over the border. The selection was made by Ctrl+Clicking the PINK layer icon in the layers panel.
There's some red noise on the grey background. It's painted manually to layer ON_GREY with a big and a small soft brush with tool blending mode = Dissolve. There's manually painted layer mask which fades the red noise smoothly.
NOTE: Noise and Dissolve are poor substitutes for real grainy textures. You can for ex. scratch some patterns onto paper with a grey pencil or other tool. After scanning them you can adjust contrast and colorize them.
Very unlikely to do that in Illustrator, however Adobe has recently released the Fresco app which is specifically aimed at producing more realistic drawing and painting textures, beyond current Illustrator/Photoshop capabilities, so you may want to try this: