I'm not sure that the kind of digital artifacts that the original poster asked about can even be called "grain." Perhaps they are inevitably less pleasing than actual film grain can be. Perhaps, post-processing using algorithms mean to imitate film grain might obscure any unsavory digitalness in them. Also, wouldn't shooting at a lower iso limit grain-like noise?
If one actually wants to acquire an appreciation of grain, I'd suggest shooting some film for a while, perhaps using a film known and appreciated for its grain, something comparable to the Tri-X of some decades ago. Any enlargement, 5x7 or above, had obvious grain. A black-and-white film with traditional grain might be best for that study.
An alternate technique might be shooting any film, including color, but experimenting with various conversions of the scans. If the viewing size is the same, cropping essentially is the same as enlarging, and at some point begins to enhance the perception of grain.
Here is an example of something I shot earlier this fall using Fujicolor 100, a reasonably fine-grained film. IFrom the color scanned file, I made a Black-and-White (colors totally desaturated) version. Then I decided to make a square crop which was also a bit shorter than the original horizontal rectangle. Part of what makes the central flower blossom pop more than the surroundings is that the sharpness of the grain, just becoming perceptible in the crop,
feels like texture on the petals and the petal edges. The uncropped version of the same image does not have this quality. Grain would seem to have a wide range of possible effects, and perhaps this could apply to the digital cousins of grain.
Zinnias.
Atlas 35 Deluxe Rangefinder, with fixed f2.8 45mm Colour Luna lens -- 1/300 sec, f/8
Fujicolor 100 film
Dev'l & scans by Dwayne's Photo
B&W conversion, cropping, etc., by me
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