Once you have the old finish off, however you accomplish that, I agree with Scorch that you can't or shouldn't start with 320 grit sandpaper. If the wood looks to be in good shape, you can start with 220, but you might just want to start with 150 grit. As you move through the grits, sanding with the grain, you'll remove the scratches from the previous grit. You can check to be sure if the scratches are removed if you'll wipe the stock with Mineral Spirits. Any scratches will show up then, and you'll know that you still have sanding to do before moving to a finer grit. As for what final grit, I used to stop at 320, but following directions from a professional woodworker that I know, this last time I went all the way through to 600 and it looked great. As you said you'd do, raise the grain a few times (at least 3 times) and sand the whiskers off with the 600 grit. As for what finish to use, that's your choice. I have plenty of finishing history with Minwax Antique Oil, and when I apply that I soak the wood on the first application until the wood won't accept any more oil, then wipe off the excess completely. The next day, I apply and wet sand with the 600 grit, then wipe the excess. I'd put on at least 4 coats, then wait a few days and then rub the stock with a medium dark paste wax applied with steel wool. The more coats of the Antique Oil you do, the more you'll fill the grain on the stock with a fine oil/sawdust mixture. Some say that when you wipe the excess oil off the stock, you should wipe across the grain so that you'll be less likely to pull that oil/sawdust mix from the pores of the wood. And if you leave the oil on the stock too long, you'll see that it gets sticky and hard to wipe off. Just rewet it a bit and it'll wipe just fine.
Now...having said all that, the pro woodworker also suggested (for indoor use) that I switch to Waterlox if I wanted an even better finish than I could get with Antique Oil. I did that on the last project and it was by far the finest finish I ever put on wood. Fantastic. And Waterlox makes an exterior version of that material. It's got 'water' in the name, but it does not contain any water and is not water-based. It's really expensive ($40 for a small can), but some day I'm going to have to try it on a gunstock. If any of you do want to try Waterlox, the first finish you put on the wood can be Danish Oil or Antique Oil. That acts as a sealer/primer. Then 4 coats or so of Waterlox. Be careful that it doesn't run, since you don't have to wipe it off like you do with the Antique Oil. You can wipe if off like you do with the Antique Oil, but you'll need more coats to fully and finally close off all the grain. The final look is like that of lacquer. Look it up on Google. There are some application videos on Utube.
Just to say it, after that first coat of Antique Oil or Waterlox, the finish looks crappy. It's only after the next coat or two that the finish starts to look really good.
The Waterlox should work great, but Antique Oil will work just fine, and is much cheaper.