Go to
Sinclair Intl. for your cleaning supplies. They stock Dewey coated rods which are arguably the best cleaning rods. Sinclair's also carries brass/bronze core brushes. Everytime I see brushes at gun shows or in stores, they are brass/bronze bristled, but have steel cores. The core should never really touch the bore, but I'd rather be safe.
I prefer the Dewey coated rod myself. I've also heard that Bore Tech coated rods are really nice too. I don't know about the carbon fiber rods myself. Stainless steel rods are common, but you have to make sure you wipe it down frequently when you use it. Aluminum and segmented rods are a no-no.
As far as the guys you see at the range who aren't using brushes, they are either (as somone already mentioned) doing the brush cleaning at home, or they have custom barrels. Benchrest shooters with custom barrels tend to avoid any brushes whatsoever because custom barrels are made such that the bore doesn't foul or develop copper deposits to the point where a brush is required.
If you have a factory tube, you will definitely need to use a brush. I don't think you can truly clean any factory barrel without using a brush. You can be running patches through the barrel for days and still have fouling.
As far as solvents go, you could use Shooter's Choice or Hoppes 9 for general lead/powder/carbon fouling. I've had better luck with Shooter's Choice though. You should also look into Butch's Bore Shine and Montana X-treme. Butch's Bore Shine advertises itself as a brushless solvent and is really popular with benchrest shooters. Montana X-treme is latest rage right now, because many people claim it is better than Butch's Bore Shine, and just as safe for the barrel. From what I understand, both Butch's Bore Shine and Montana X-treme do have Ammonium Salts in the solvent. I was told ammonia could cause pitting, but apparently, these two solvents don't have that danger. As far as copper solvent, I would use Butch's Bore Shine or Montana X-treme before I'd go with Sweet's. Sweet's 7.62 is a very good solvent in that it will get the copper out fast and efficiently, but it will pit the bore if you keep it in too long. I only use Sweet's every few hundred rounds and I make sure that after I run patches with Sweet's through, I clean dry the bore completely and run and oil through it to make sure any Sweet's is gone. Some people use 99% isopropyl alcohol on a patch after running copper solvent to make sure any solvent left in the bore is deactivated.
When I clean my Savage 10FP, I first run 3-5 patches saturated in Shooter's Choice solvent (regular, not the copper solvent) through the bore to get the carbon gunk out. I'll let that sit for a few minutes and run a dry patch through it (note: Always push the patch in one way, chamber to muzzle, and remove the patch from the muzzle end. Don't push the patch forward, and pull it back.). Then I'll wet a brush with Shooter's Choice and run that through the barrel back and forth a number of times, depending on how many rounds were fired between cleanings. After that I'll run patches saturated in Shooter's Choice through the bore to get the loosened fouling out. At this point, it only takes a few patches before the blue starts to show on the patches, and the black fouling starts to go away. When I think the lead, powder, carbon fouling is pretty much gone, I'll run a few dry patches through the bore. Then I'll take the bore guide out and clean the chamber. With a .308, I use a .45 brush with a patch wrapped around it and push it into the chamber to wipe it out. After I clean the chamber and let the bore dry for about 10 minutes, I'll run a dry patch through again and make sure there's no more solvent left.
Depending on how many rounds I've put through it, I'll then run a copper solvent like Butch's Bore Shine or Sweet's. Butch's Bore Shine is actually an all around solvent, but I've found it to be a good copper solvent after the barrel has been cleaned of the carbon fouling, and it's safer to leave in the barrel than Sweet's. I usually run several patches wet with the respective solvent through the bore until the blue starts getting less noticable. Then I'll run dry patches through to make sure it's dry.
After this, I just go back to shooting, or run a patch wet with a non-Teflon gun oil through the bore, and follow it with a dry patch.
A good tip is to wipe the rod every time you run it through the bore. Even if the rod is coated, fouling can stick to it and it may get rubbed against the bore as you run the rod through.
Rifle cleaning is a religion because there are so many different methodologies. You should definitely get (1) a quality rod like a Dewey, (2) a decent bore guide, (3) brass/bronze only jags and brushes [no stainless steel brushes, ever!], and (4) quality patches that fit your jag and bore.
A decent rifle cleaning rest is a good idea too if you ever want to clean at home or at the designated cleaning area off the firing line when at the range.
Good luck and good shooting.