I've attached a reference diagram from the Sears parts website that I wish I would have had before I started. It's a little grainy, so I modified it so you can actually see the parts names and numbers. If you aren't sure you can do this sort of project, please know I had never even taken a vise apart before. All steps were very straightforward minus the removal and replacement of the collar, but it can be done without specialty tools.
Step 1: Disassembly
Before you begin, get an empty box handy for parts storage.
Start by loosening and removing the lock nut handle (#11 - aka the little handle on the side of the vise). This handle can be seen in the first picture.
Next, completely separate the front and back jaws by loosening the main vise screw/handle until the two jaws are disconnected.
Then, remove the four face screws holding on the two jaw face plates. I would suggest cleaning out the slots on the screws with wire brush and/or WD-40 or your preferred solvent, so that the screwdriver has good contact with the screws. Also, if the screws are really tight (they will be), use a screwdriver with a wrench assist as pictured.
Next, remove the center bolt that holds the swivel plate onto the back jaw. Holding the back jaw upside down, it is the big bolt in the middle of the swivel plate.
Now you can remove the vise nut. It is the thing that the center bolt was screwed into (inside back jaw). To remove the vise nut, push it toward the back opening of the back jaw. You might have to knock it loose with a wooden dowel.
Now the tricky step: removing the collar. The collar is the little metal ring that holds the screw handle onto the front jaw (you can see it at the top of the picture showing the screw handle resting in the front jaw housing). I finally got it off by prying the slot in the collar wider, so that it would allow the screw/handle to to slide out of the front jaw housing. I used a couple flathead screwdrivers to pry it off, but it took a lot of experimenting.
Your vise is now disassembled.
Step 2: Degrease and Remove Paint
After using some spray solvent and wiping off all the old grease and dirt I could reach, I took a wire brush and went to work. I did the back jaw completely by hand, which I would not recommend unless you enjoy that sort of thing. I don't. So I switched to an angle grinder with a wire brush. Make sure and wear eye protection and I would also recommend a work apron and gloves. Let the grinder do the work. Don't push really hard, because you don't need to.
I touched up the small areas the hand grinder could not reach with a Dremel tool and small wire brush. Don't use a grinding stone bit. It will dig into the metal. Having access to another small vise is really helpful with the small pieces.
Clean down to the bare metal.
Step 3: Mask Surfaces and Paint
Using painters tape, mask off the areas where the jaw faces mount, the little flat anvil section on the back jaw, and the thread on the center and locknut bolts. I used foam hearing protection plugs to protect the threads inside the vice nut and areas where the center bolt passes through.
After researching which colors best matched "Craftsman Red", I picked Rust-Oleum Regal Red (one 12 oz. spray paint can was enough). Follow the directions on the can, but when in doubt, multiple light coats is best.
When the paint is completely dry. Remove the tape.
I left the screw/handle and lock nut handle unpainted.
Step 4: Lubricate Key Components and Reassemble
After researching the best thing to use for a lubricant, I decided to use Frog Lube Paste. Most people will probably prefer white lithium grease, which, of course, is fine. I lightly coated any vise surface that rubs against any other vise surface. The screw handle threads, top of the swivel base, and front jaw sleeve (the part that inserts into the back jaw) got a a slightly heavier coating.
Assembly is essentially disassembly in reverse, with one exception: the collar. Putting it back on and securing it was even more fun than taking it off. Tightening the collar on the screw shaft takes some experimentation. I ended up moving it into place and used a plastic wedge between the inside of the sleeve and the collar to hold it in place. Using a flat blade screwdriver, I pried/pushed it around the screw shaft until it couldn't slide off its slotted indentation. Easier said than done, but it is possible.
Once the vise is assembled, work all moving parts to ensure proper function. Wipe off any excess lubricant.
Step 5: Secure on Workbench
Completely unnecessary step: I elevated the vise just a bit by mounting it on an old scrap piece of walnut that I cut and routered to match the swivel base. It allowed me to use an old weird-shaped piece of wood that wasn't doing anything else, plus I like how it looks.
Enjoy your new vise.