Okay oil finish:
There's a few kinds of oil one can use. Linseed oil is one of the first to come to mind, but I prefer tung oil for a finish, due to its waterproofing properties. You can buy pure tung oil and cut it with mineral spirits if you want to go the traditional route. I've been using Behr "Tung oil finish", which is a mixture of solvents and tung oil. Tung oil by itself is a great protector, but takes a long time to cure. Cutting it with different solvents helps to make for a faster curing time.
But enough technical nonsense. Once you've got your rifle stained to the colour you want, give it a quick rub down with some varsol to get rid of any dust or grease residue. Take a rag and dab it in the oil. Just like with the stain, start to slather the wood in oil, quickly and efficiently. Put on a heavy coat. The wood will start to drink the oil right up. Keep adding a bit of oil to your rag, and keep rubbing oil in. The wood will keep absorbing the oil. Eventually, you'll notice that the wood is starting to drip the oil, and it just stays on top of the surface. This means the wood has absorbed as much oil as it can (for the moment). Let the wood sit for ten minutes, then wipe it down to remove the excess oil.
Hit the stock with heat from a heat gun/hair drier. (Wear a mask) You'll see the oil soak into the wood even more, and you'll see little beads of oil on top of the wood. Give the wood another wipe down with a clean rag, and let it sit for an hour.
After an hour, come back, and briskly hit the stock with steel wool. You want to raise up a slight amount of "dust". This dust will be a weird mix of wood dust and finish. Keeping this mix of dust on the wood, give the stock a light slather of oil. The "dust" will serve to fill in any grain depressions and will help seal the wood. After this, hit the stock with another heavy coat of oil, watching again for the wood soaking up oil, and then reaching the point where it can't take any more. Wipe it down, wait ten minutes, hit with the hair drier, wipe down exess, let sit for an hour.
Repeat the oiling one more time, no longer using the steel wool. I do 3 coats and then let the wood sit overnight. I do this for about 3 nights, and then let the stock sit in a corner for a week, giving it a good chance to air out.
After that, I make up a mix of 1 part tung oil, 1 part mineral spirits, and 1 part furniture wax. I put the wax in a shoe polish tin, heat it over an ELECTRIC HEATER (If you use a propane torch, you'll light the mix on fire and have an exciting evening. Trust me.) Once the wax is melted, I add the mineral spirits and the tung oil, then allow the mixture to cool. It will turn into a weird sort of jellied mix that smells really bad. I call it stock "polish".
Using a clean terry cloth rag, I dab a corner in the mixture, and rub a moderate amount on the stock. You will see the stock start to look a bit glossy. Let the polish sit for half an hour, then add another coat or two. Let sit for another 30 min, then buff the stock. The stock will start to take a shine.
Add a coat of this stuff once/day for a week, and at the end of the week you'll have one heck of a nice looking rifle. The wood will take about 14 days after your last coat of polish for all the solvents and funky smells to evaporate. After that, you'll be good to go. Maybe add a light coat of tung oil every 3 months to keep the stock shiney, and that's all you have to do. And if it ever gets scratched, just add a coat or two of "polish", buff, and that's it.
*edit: Oh and I noticed you have checkering on your stock. Don't hit the checkering with sandpaper: it will make a mess of the checkering.*