I use these items for cleaning my 22 rimfire barrels. I also say why above.
36" 17 cal Dewey coated rod. If you get a shorter rod and someday have a 28" barrel rifle you may not have enough rod if you use a bore guide. This is a Dewey model 17C-36.
For patches I have been using these Pro-Shot patches for a long time and they work perfect around the Parker Hale jag. Next size down works great for a 17 rimfire, I have those too.
Jags, I have tried several throughout the years and have settled on Parker Hale as the best for me. With the patches wrapped correctly it makes for a tight fit. This particular nylon brush is made my Tipton, IMO it doesn't do a lot of cleaning but I believe it's a little more aggressive than the patch alone plus it shouldn't disrupt your seasoning in the barrel like a bronze brush will. It is good for attempting to remove a carbon ring because you can easily reverse the direction while in the chamber, something that is hard to do with a bronze brush. More on that carbon ring at the end.
These Dewey bronze brushes are the only ones I ever use. They do wear down so I buy a pack of them at a time.
An adapter is needed to convert the 17 cal to 22 cal attachments. This is a 17A adapter found on Dewey website.
There's probably thousands of cleaning solutions on the market. I prefer a 50/50 mix of Kroil and Shooters Choice MC #7.
For quick removal of a heavily built up carbon ring I use this. You can push the JB into the felt and saturate it. Some will wipe off but most will remain for a good cleaning. I use a really old 22 cal rod for this step. If you use your 17 cal rod and this VFG felt adapter you will need another adapter called a 22A from Dewey website.
As for the infamous carbon ring, I am not convinced a carbon ring has any effect "good or bad" on accuracy. What it will hurt is reliability, I know this first hand with extraction issues.
If you notice one of my rods has its coating scraped off pretty bad. This is a direct result of not using a bore guide. Let the results of my mistake be used to your advantage by showing you first hand the damage that can be done if not using a bore guide.
In the end everyone has their own cleaning methods and products used. I would say except for two things nothing is right or wrong, it just is.
Those two things are to never clean from the muzzle end unless it's your only option. Never use a steel brush. Sorry for the lengthy reply, sometimes I go overboard with the simplest of things.