Looks like a nice vintage/old piece. Brush it off with a wire brush, not too hard of scrubbing. Apply a stripper to the top, especially the painted areas, and on some of those underside areas, as well. It should clean up fairly well. You want to maintain the old look, but not the off-colored stuff and the paint/stain (?) debris/contaminate blemishes. I'd say not to try to strip it to be perfectly clean and free of old patina, i.e., leave some vestige of age and use on it.
On the first and last pic there is a "clean looking board"... looks like a relatively new common 2X4... a past repair, replacement of that board?.... I can't tell for sure. It just seems a different color, compared to the other boards. That board doesn't seem to match the other old looking boards. If possible, replace that board with an old salvaged appropriate looking board. If replacing is not an option, then soak a Brillo (scrub pad) size wad of steel wool in 2 cups of vinegar for a few-3-4 days. Once the steel wool has dissolved, dab a small amount (a couple of Q-Tip dabs) of the solution onto an inconspicuous area of that odd looking board. Once dry (5-10 minutes in warm/hot sunlight), the solution will faux age the wood, possibly "coloring" it to a more appropriate color match to the rest of the wood. If you like this test result, then you can faux age the rest of the board.
*You can test the faux aging technique on some scrap lumber, to see how it dries, colors, etc. It paints on like painting with water. With a small or 3" brush, wet the surface fairly good with full brush strokes (drips and runs are ok). Once a test area (1 sqaure foot?) has been wet, wipe off the excess with paper towel. Once you see the process of wetting and wiping excess (it goes fast), then you can brush on/wet larger areas. For that table, I could/would wet probably 1/4 the surface, before wiping the excess. , then move on. When wetting, you can slap the solution on really fast, do the quick wipe down, and move on. But test a scrap board first, to get the hang of it, if you opt for this option/technique. It's cheap and easy (a neat trick) with good results, for many "aging" applications.
A lacquer or shellac finish would be ok, but if you plan to use (or family abuse) the table, then a more durable finish would be recommended. But lacquer and shellac repairs easily. You can apply just about any finish over a shellac oat, also, should you later decide to put a more durable finish on it.... i.e., with a shellac finish, lightly sand, then apply a different finish.
If you strip the table and apply a lacquer finish and you end up with some spots that "aren't right" (not smooth, meaning defective resulting coat, then I'd guess all the old paint (or any other old finish, if applicable) has not been completely removed. The solvent of the lacquer (acetone) can act as a stripper (to some degree). If any old finish/paint is still on the table, the acetone might cause it to surface, to rise up from the wood. Sanding and re-applying another coat or two should fix that. Try to make sure all old finish is stripped off, if you do some stripping.
If your table top comes out nice and fairly clean, then you might only need is to wax it, rather than applying a dedicated finish. Lightly sand the surface smooth, somewhat smooth, then wax. Just a waxed look is a very nice alternative to finishing, especially for an old vintage/antique raw-wood (preference) farm table, farm-like table. BreWax or TreWax are good waxes for this application. Read the instructions on the cans. A more expensive wax is Pate Dugay, which you can get in colored selections. Pate Dugay takes a good bit longer the dry (maybe a day or so longer, depending on the raw wood and how much paste you apply. Don't apply big gobs of Pate Dugay. Do small areas at a time. Pate Dugay results are very nice, just takes a lot longer for application, drying, buffing, what worth the effort. Alas, Pate Dugay is somewhat expensive.... Takes years to use up a whole can.https://www.google.com/search?q=pate+du ... e&ie=UTF-8
No matter what the project, prep work is important. Take time to strip, clean or whatever else, before finishing.... including hardware inspection, cleaning, etc. Your results will be its best if care is taken with the prep work. I think that table is worth the effort, it's a nice table.