I picked up a used (rather heavily) Franchi Black Black Magic.
The metal isn't bad, and the action is super smooth so I'm focusing my attention first on the stock which is rough.

I have basic wood working skills from wood shop and understand starting with low grit, stay with the grain and work up.
Issue is there is a good amount of checkering. Checkering is way beyond my skills and tools, and honestly it doesn't look that bad.

As of now my plan is to mask off the checkering, sand around, and refinish with gunstock oil. I'm just not sure if it'll match after I'm done, and how it would look.

I can't really come up with any ways to remove finish from the checkering without destroying the checkering.

Any ideas?

As I'm setting this up as a a skeet gun, I've also considered applying a grippy type paint to the checkering. Not sure how that would come out either.

ETA I should specify, my wife's side of the family and their friends are well off, and enjoy skeet shooting. I enjoy it, but not enough to drop a few thousand on fancy guns.
Picked up this gun from Sarco, and plan to refinish it to make a nice looking, functional trap gun. I'll prob also use it for bird of I ever go too.

Strip as much Finish as possible using a stiff nylon brush. Buy a checkering "pointing" tool to clean the bottom of the grooves. When the time to refinish comes, mask off the checkering with duct tape. Finish the smooth part of t the stock to your liking, use as many coats as you need to. I like Minwax Antique Oil Finish. Unmask the checkering and either use a non polymerizing oil like fret board oil which gives a darker flat finish. Or you can cut the Minwax by a third with mineral spirits and brush in sparingly. Wipe out any excess oil.

Should I use any solvent with the brush?

I'm sorry, stripper such as Citri-strip. My brain outran my typing. You might be able to get checkering grooves clean with a tooth pick. I use a pointer because I have one and I tend to be kind of OCD. PM me if you need to.

As wingsnthings said..
Chase the checkering with the correct checkering tool. You will probably need at least 2. One border tool and one w shaped.
The real bitch is holding the stock so you can rotate it and be comfortable as you chase the curves in the grip area.
A stock jig helps tremendously.

Yes, use zip strip or other..?
The goal is to not tear up the tops of the checkering. So..checkered areas get special attention and care.
Brush the zip strip on and then use a nylon brush to gently clean it off. Probably need to do twice.

Trying to avoid getting any special checkering tools.
The lines are pretty good.
Will a good course nylon brush and the solvent you recommended be enough? Or do I need the tools?

I do have a wood working store near my office. Guess I'll have to take a look.

Citri-strip is a great product. I used it to refinish an old shotgun recently. I’d give that a try with a nylon brush first. It’ll likely take a couple applications. You have to to use a rigid tool to scrape the valleys of the checkering. If you don’t want to purchase tools, you may be able to find something plastic around the house to match the contour.

You can grind or file the tip of a scribe or even a nail to closely match the checkering.

I've used brass bristle brushes to clean out checkering. Let the stripper do the heavy lifting. Look for a stripper with methylene chloride in it for heavy finish coats, otherwise citri-strip.

When you sand the stock, lightly wet the wood and let it raise the grain and sand it smooth. Repeat until the grain stops raising.