Just offering a little bit of Dream Analysis for you!
Being that you are dreaming of your cat (which is important to you), you are conscience of how much they mean to you. Being that you heard that it was good to put him in the dryer and that you actually tried it (though, by putting him in his cage before doing so eludes that you KNEW it would hurt/harm and you were 'protecting' him by securing him in his cage). Your curiosity was piqued by wanting to see if what people were saying was true (hence actually putting your cat into the dryer) and you were following advice [here is where I think your trip to the vet comes into play; you took him to the vet (following the given advice for his shots) and felt as if you couldn't protect him from the pain (which is why, in your dream, you had put him in his cage before
putting him in the dryer)].
After you had turned it on, you heard him cry and you knew it was scaring/harming/hurting him, so you turned it off and consoled him. This is your protective nature coming through in the dream because you couldn't well take him from the veternarian after the first shot!
By promising him that you'd never do it again, you feel as if you have learned your lesson about following given advice.
Now what you said about 'I must have been sad about causing my baby pain...' in your dream, you tried to do things a little differently by taking away the pain (taking him out of the dryer).
The significance of the dryer you ask? Here it is: you dry something after you wash it... why do you wash something? To clean it, or get rid of the dirt. Dirt is another way of saying something that isn't comfortable or unwanted negative feelings. You put your cat into the dryer because you wanted to make up for the 'guilt' (that dirty feeling, still lingering in the sodden clothes) for taking him to the vet.
You're dreams do make sense -- you just need to make sense of them.