After soaking the parts for bit, look them over and make sure the cosmo is out of the nooks and crannies (Q-tips work great!).
Once you have them out, wipe them down and let air dry. Once dry, clean and lube them as you would after shooting (Hoppes #9, CLP, whatever), then oil accordingly.
The only thing I didn't like about leaving my stock in the form I got it was this: If you go shooting, the gun will heat up and leach cosmoline all over YOU, your clothes, and any gear you have close by. I love my rifles, but I'm not gonna trash my clothes for it.
It will be most noticeable along the warm metal parts, but the butt of the barrel will be very close to your clothes.
If you use the Turpentine method I described earlier, it will take 3-6 wipedowns to really get the cosmo out (though some may still be in the DEEP parts of the rifle (not worth worrying about at this point though), but it will dry quickly for you to apply the wax I mentioned earlier or something like Boiled Linseed Oil or your favorite wood treatment.
Previously, I had used Howard's Restor-a-finish to do the wood, then applied their orange oil/beeswax mix to it and it looked okay for a while. I just preferred the Tom's mix, especially after seeing my Garand's results.