I haven't refinished speakers, but I do the occasional piece of furniture.
Treat it like an antique. Don't take a power sander to it. You'll cut through the veneer pretty quickly with a power tool.
Start off with a chemical stripper. You will need two coats at least. I would suggest that you do not use the regular all purpose full strength stripper. The hardware stores will also have a somewhat milder stripper for antiques. Use the milder stripper. It will take longer but you'll have more control over it as it's slower acting. I would also suggest that you use a gel, rather than a liquid. It's easier to work with, especially since you will have vertical surfaces to strip. It wouldn't hurt to take your time and do one side at a time. There's no rush. Again, you will have greater control. If there is still some finish on the wood after two coats, you can treat troublesome spots with a touch of stripper, or sand by hand. Try not to use a coarser sandpaper than 120 in the stripping stage. It's extra work, but again you want to be careful not to penetrate the veneer. Don't worry about the colour. You will be surprised at how bleached it will turn out, even if you started with a dark colour. After stripping, wipe down with a paper towel or cloth damp with a petroleum ditillate. (paint thinner or varsol, or gaoline for that matter, they're all the same thing). This is necessary to deactivate the stripper. Next, sand by hand, not with power tools. But use a sanding block, not your fingers. Unless you're careful, using fingers under sand paper will groove the surface. Sand with a fine sandpaper, say around a 200 grit. Again it takes longer but the objective is to do a good job, not a fast job. Besides, it's already probably pretty smooth, so you don't need an agressive sandpaper. After sanding, again wipe down with a petroleum distillate. Now you're ready to stain and urethane. Just follow the directions on the can. Wipe the stain on with a lint free cloth or a foam brush. Unlike stripping, try to stain all at once, and do both speakers at once. Keep the stain on for the same length of time. This will facilitate consistency of colour. Consistency of colour is also facilitated if you use a conditioner on the wood immediately before staining. This is a necessity if you are using softwood, but it's good to use it for all types of wood. When you urethane, it is is much better to use multiple thin coats, rather than fewer thick coats. You should do at least three coats, with the first coat thinned as per the directions on the can. Typically it's a 50/50 mix of urethane and solvent. Lightly sand and wipe down between coats with a very fine sandpaper. The objective here is simply to roughen the surface of the previous coat of urethane so that the next coat has something to grip onto. You should not be removing the coat that you sand. Use a foam brush for the urethane rather than a bristle brush unless you're experienced at doing this. Be careful stirring the urethane and using the foam brush as you want to avoid bubbles. Like staining (and unlike stripping), you should urethane it all at once.
I doubt if there's is anything I said that you can't find in a do it yourself brochure or book from the hardware store.
However, there is one more very important point that do it yourself books from the hardware store won't tell you.
When you refinish, take the drivers out of the speaker!!!All of the chemicals: the stripper, solvent, stain and urethane could (will) damage the cones if they touch them. Even fumes from these products can cause damage. So take the drivers out. You could try unscrewing them and putting them in an air tight plastic bag without removing them but I think it's probably easier and better just to get them totally out of the way.
Treat it like an antique. Don't take a power sander to it. You'll cut through the veneer pretty quickly with a power tool.
Start off with a chemical stripper. You will need two coats at least. I would suggest that you do not use the regular all purpose full strength stripper. The hardware stores will also have a somewhat milder stripper for antiques. Use the milder stripper. It will take longer but you'll have more control over it as it's slower acting. I would also suggest that you use a gel, rather than a liquid. It's easier to work with, especially since you will have vertical surfaces to strip. It wouldn't hurt to take your time and do one side at a time. There's no rush. Again, you will have greater control. If there is still some finish on the wood after two coats, you can treat troublesome spots with a touch of stripper, or sand by hand. Try not to use a coarser sandpaper than 120 in the stripping stage. It's extra work, but again you want to be careful not to penetrate the veneer. Don't worry about the colour. You will be surprised at how bleached it will turn out, even if you started with a dark colour. After stripping, wipe down with a paper towel or cloth damp with a petroleum ditillate. (paint thinner or varsol, or gaoline for that matter, they're all the same thing). This is necessary to deactivate the stripper. Next, sand by hand, not with power tools. But use a sanding block, not your fingers. Unless you're careful, using fingers under sand paper will groove the surface. Sand with a fine sandpaper, say around a 200 grit. Again it takes longer but the objective is to do a good job, not a fast job. Besides, it's already probably pretty smooth, so you don't need an agressive sandpaper. After sanding, again wipe down with a petroleum distillate. Now you're ready to stain and urethane. Just follow the directions on the can. Wipe the stain on with a lint free cloth or a foam brush. Unlike stripping, try to stain all at once, and do both speakers at once. Keep the stain on for the same length of time. This will facilitate consistency of colour. Consistency of colour is also facilitated if you use a conditioner on the wood immediately before staining. This is a necessity if you are using softwood, but it's good to use it for all types of wood. When you urethane, it is is much better to use multiple thin coats, rather than fewer thick coats. You should do at least three coats, with the first coat thinned as per the directions on the can. Typically it's a 50/50 mix of urethane and solvent. Lightly sand and wipe down between coats with a very fine sandpaper. The objective here is simply to roughen the surface of the previous coat of urethane so that the next coat has something to grip onto. You should not be removing the coat that you sand. Use a foam brush for the urethane rather than a bristle brush unless you're experienced at doing this. Be careful stirring the urethane and using the foam brush as you want to avoid bubbles. Like staining (and unlike stripping), you should urethane it all at once.
I doubt if there's is anything I said that you can't find in a do it yourself brochure or book from the hardware store.
However, there is one more very important point that do it yourself books from the hardware store won't tell you.
When you refinish, take the drivers out of the speaker!!!All of the chemicals: the stripper, solvent, stain and urethane could (will) damage the cones if they touch them. Even fumes from these products can cause damage. So take the drivers out. You could try unscrewing them and putting them in an air tight plastic bag without removing them but I think it's probably easier and better just to get them totally out of the way.