The key quote from Tolkien that answers this is contained in Letter 246, with my added emphasis:
It was part of the essential deceit of the Ring to fill minds with imaginations of supreme power.
This implies that the Ring is useless to anyone but Sauron, but it tempts you into thinking it's a source of power, and in that way it gets you to wear it, and so gains control over you.
[Source: an answer to Would the One Ring even work for anyone but Sauron?]
Basically, whenever someone wears the One Ring, they 'change': for example, Sméagol became Gollum, and even Frodo started showing behavioral changes due to the One Ring gaining control over them as stated in the answer.
So how exactly does the One Ring affect the wearer?
Here, I mean to ask if it's ever told by a character in the books/movies the changes that the One Ring brings to the wearer. For example, it corrupted Frodo and he forgot his friendship with Sam. Similarly, it also affected Sméagol.
Beside these changes, are there any other changes that the wearer of the One Ring goes through? (For example, has a Ring wearer ever described how wearing the On Ring 'changed' them, i.e., they felt different after wearing the One Ring?)
As clarified in the comments, I have not read the books but I have watched the movies, nevertheless I will accept answers from the books too.