James,
This is a link to Eric Uptagrafft's instructions for how to clean a .22 rimfire:
http://www.uptagrafftllc.com/tech-blog.html
In case you don't know him: Eric is a longtime member of the U.S. Army Marksmanship Unit and two-time Olympian. I have shot with him several times and, judging from his scores, his cleaning methods seem to work pretty well.
Here's another take on cleaning equipment and methodology from another rimfire expert:
http://www.rrdvegas.com/rimfire-cleaning.html. It's long, but a good read.
I use a .22 Dewey coated rod. I can't say it's better than the others recommended above but it works fine.
The main thing about the jag and patch is to find the right combination of jag and patch size to get the right fit in the bore -- more difficult, in my experience, than with centerfire rifles. I use a pierce-style jag similar to the Pro-Shot with Sinclair 1 1/8" patches. I also use a Parker-Hale wraparound type jag (bought it from Champion's Choice) with Sinclair 1 1/4" patches. Both combinations work well.
As far as a rod guide, I use and recommend this one:
http://www.mwerksllc.com/boreguides.php. Jeff makes them really well and you'll need to know which brand of rod and the exact rod diameter when you order. I called Jeff before I ordered to make sure I did it right. Might be worthwhile to call him before you get a rod to get his recommendation.
Dave Rabin