If you're going to do what I'll show, you only need one file, but you do need abrasive paper and float glass. I don't know how perfect the flat surface has to be. Mine is very close to it (perhaps not able to slip a .0015" feeler between an undamaged starrett 24" straight edge and any point on the surface, and the support for me is 4 1/8" of ash and I work over a leg, so it may stay relatively close to that. I'm sure that level of flatness isn't required, though). I also have two granite reference surfaces, but I never need them.
I do have marking fluid, but someone could use a magic marker or something easy. If one already has the straight edge and the lapping surface, all you really need is a file that looks like the one I attached here (this is a simonds black maxi.), and a file card.
https://s2.postimg.org/3l34v9wcp/s-l1600.jpg
The challenge with lapping only is that you will have a very hard time doing much metal removal and still be within a thousandth or two. You can certainly make a functional plane, though. The challenge is several multiples greater when you move to infills and start trying to lap mild steel instead of cast. It laps so slow that it's untenable, and draw filing (or filing in general, maybe not specifically draw filing) becomes the only reasonable hand method if there is more than a little to remove. Plus, you may pein everything together and find the sole proud a hundredth in the middle, which is a horrible situation for hand lapping, as you have to remove that hump and you may still have some cosmetic stuff on the ends to get through. I have no idea how long it would take, but that was the situation on my last infill and I wasn't willing to find out after I blew out one application of 80 grit PSA roll in about 5 minutes and didn't get very far. I would've also ended with a plane that had proud ends, and I didn't want to do that.
Here are the steps to what I do:
* examine the sole of the plane with a straight edge (a really good one that is known to be super straight). If you have a good lap, you can start by running the plane down the lap a few times with coarse paper, it will mark the high spots.
* If the plane is high on the ends (ends touch a flat surface, but not the mouth) and made of cast iron, you can continue to lap if you'd like. You'll end up with a very flat plane if you stop at the right point, as long as the plane isn't coffin shape (coffin shape metal planes tend to end up with the sides relieved as well as the ends because there isn't enough forward and rearward metal, I guess, to keep you from accidentally dubbing the sides). If you overshoot, then you can go to filing below.
* file the high spots with said file above by applying the file diagonally on the surface of the plane where the high spots are, but not overlapping the edge and with gentle care if crossing the mouth - no crossing the mouth if the file is pinned, and no harsh work at the mouth even if it isn't. The way to do this is to put fingers on top of the end of the file with the non-dominant hand and pull up on the handle with the non-dominant hand a little bit and push in short strokes. In doing this, you are flexing the file a bit and quasi draw filing with the first inch or two of the file only, never filing on the outside perimeter of the sole, but staying within it. As an amount of swarf builds up on the plane, stop and brush it off, and check the file for pinning. The files I showed above will flex slightly and you can quickly file material without filing sides or edges of the plane sole, and the file has teeth all the way to the end (or close to it), so it's not just rubbing the surface. They come with a handle here for 8 bucks and I haven't worn one out yet. They are also great for draw filing elsewhere because the broken tooth pattern is fast for rough work, but will still draw file.
* stop from time to time and lap the plane again. Lap until the filed area disappears. The high spots should be less high, but still should be discernible from spots that are either not yet lapped or that have been filed and are not being contacted by the lap.
* continue the filing process, filing in the center (inside the perimeter of the plane on the high spots if they are somewhere other than the center) and lapping to bring down the edges to the place where you filed being careful not to over file the center. As you get close to flat, the filing done each iteration should be less and less so that you don't run the risk of leaving a large amount to lap out. that is, if you think you're taking a thousandth off with the file or two thousandths, as you get to the end of the process, you should only remove a fraction of that and then lap the work out, remove a fraction, then lap the work out, etc. As you approach getting a lap that covers the whole bottom, the entire plane should be flat. If it is not, continue iterations until it is.
There are two points of care that have to be taken:
* If the file pins, you have to stop right away and card it (for the newbies - that means if it clogs, you have to use a special stiff wire brush, a "card", and get the pinning out) . Pinning will leave a deep scratch that you have to work out. If you are only working on a high spot, it's not a huge deal but:
* care must also be taken to make sure that you don't run a pinned file over the mouth of a plane. The pinning that is deep in an expanse of metal will be large dents at the edges going over the side of the mouth at the back and into the wear at the front. They'll never come out unless you still have some mouth that you need to file away (if making an infill, this should be done before the mouth is filed to finished size). Even then, the damage may be deep enough at the drop offs that you don't get it all out.
* I guess there is a third. The file should work inside the perimeter of the plane only and the lap finish the work to consolidate it all. If the file is allowed to overrun the ends, a low spot will almost certainly be created somewhere.
I don't know if that's clear or not. The file and the back and forth are the nuance. I am sure that the average person can do better than 1 1/2 thousandths with this, because I can. It's also something that on a pretty horrible smoother sole made of mild steel, you can do all of the work in about an hour. A machinist won't approve, but machinists can take a flying leap through a rolling donut if they think they are experts about what makes a good woodworking tool.
I never lap with anything higher in grit than 80 grit. It's a waste of time. I intentionally finish the process with worn paper that is cutting slightly slow, and if it's not cutting slow enough, I find another plane and lap it for a little bit to take the initial snot out of the aggressive paper. The file is doing the work, and the lap is only doing a little. If I think I need a better surface finish than worn 80 grit paper, then I lay a piece of 220 wet and dry on top of my 80 grit lap and use it only enough to make the surface a little more refined, and then after that, a wisp of scotch brite and a careful cleaning of the metal with vigorous brushing.
Include a file card in the cost of this if it's not had. I don't do this for all cast iron planes because they lap quickly and you can get very good functional flatness, unless they are already a banana to start. I may start doing this on long planes if I buy any in the future, because they are hard to lap unless they start with their ends low (mouth proud of the surface).
Nobody needs the feelers, I was just curious about what I was attaining. I wouldn't use cheap feelers, either. They're not reliable, and good ones are not expensive. I also wouldn't use a cheap straight edge if you are talking about a couple of thousandths - starrett is it, and they're not cheap, but a 24" starrett 380 won't be the most expensive thing in the shop, either.
This is actually a very easy process, the only thing that you end up doing is sweating a little bit. I'll still make a video of it. Other people should know how to do it, it's cheap, and knowledge is free.
The ability to square is pretty much the same thing, mark the high spots after the sole is flat and file them off until everything is close enough to finish (cosmetically) with a lap on the 80 grit. the work after that for cosmetics on the sides can be done carefully with a small block and sandpaper, being careful not to do more than just improve the cosmetics - no need to overdo it and sand a bunch of additional material off.