Go to Wally-World. I went and bought a deluxe cleaning kit that came in a nice wooden case. It has gun yokes and plenty of room to store other items. I also bought some nitro gun solvent, some extra patches, and some gun oil.

The kit can handle rifle and shotgun.

The step I take are -

1. Bore brushing several times to loosen up the build up
2. Gun solvent several times
3. Run a dry patch through the weapon until it comes out fairly clean
(you will never get a completely clean patch out of it)
4. Use a good gun oil on a patch and run it through several times
(DON'T soak the barrel/bore with gun oil - A light coating is all you need!)
5. Run a dry patch back through once.
6. Take a soft bristle tooth brush and gentle go over the outside.
7. Can of compressed air will help to blow away the dust and build up in the outside nooks and crannies.
8. Wipe down the weapon with a good silicone impregnated gun cloth.

Bolt Cleaning on the Rifle -
1. The bolt should be removed before you clean the rest of the rifle
2. Take the tooth brush and gentle clean the entire bolt (DO NOT disassemble the bolt unless you are a professional gunsmith)
3. Wipe the bolt with a gun clothe

Shot gun cleaning is similar to the above steps except for the bolt remove (unless it is a bolt action shotgun)
Besure to NOT to heavily oil the slides on an autoloding shotgun. I very light coating wil suffice.