Re-finishing a cedar chest

Have a cedar chest that I'm looking to touch up. Picked this up at an antique mall, interior is immaculate but the exterior has some blemishes that I'd like to work on.
What I didn't realize with the chest at first was the exterior, at least the top has a veneer that wraps around. Have never worked on veneer surfaces but I'm guessing due to how thin they are there may be a whole different set of rules of how to approach re-finishing.
The feet on this don't appear to have a veneer layer, so that I'm guessing may be somewhat easily tackled with a bit of sanding to smooth out the scratches/gouges.
If folks have a minute to give some pointers.

Thanks!
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Pictures are included.

As with any "re-finishing" project it's best to take the least destructive path first and see if the results are satisfactory. That means starting with a thorough cleaning as well as testing to ascertain what the current finish actually is. A little DNA in an inconspicuous spot can tell you if it's shellac. If no, try a little lacquer thinner in the inconspicuous spot to see if that's what's on there. If not, it's likely some form of varnish. Shellac can often be reconditioned with DNA and/or wiping on a thin coat or two. Lacquer can be similarly reconditioned in many cases. If there is a lot of damage or it's varnish which is harder to "repair" rather than replace, use of a chemical stripper may be indicated...and that avoids most sanding that poses a risk for veneers. If it's varnish, you may still be able to repair minor dings and re-coat to freshen it up. Regardless of what needs to be done, avoid sanding as much as possible. Use a sharp scraper to lightly deal with rough spots when you can and very limited sanding abrasive only if necessary --try finer paper first so it only cuts the rough and not what's below it and use a block when possible.

IE...go slow and only do what you have to to get the piece where you want it to be after you recondition it.

BTW, for the interior which is bare wood, a light sanding will "freshen" the cedar lining up...again, finer paper for that.

Thanks Jim!
Some of the marks on the lid appear to be beneath the finish, almost like water marks. Aside from the pictures I hadn't looked it over in that much detail to say for sure. One of the projects that will be coming up, just not on the front burner yet.
One thing I saw but hadn't mentioned, between the various possible finishes, does one tend to show smaller cracks (have to get right down and stare at it little) around corners? I probably won't be able to start in on this for a few weeks, but this tidbit of info might help simply in planning for the approach.