A number of thoughts:
I myself would choose to make this of solid wood, and to use joinery such that minimal fasteners will have to be used and concealed. This involves considerably more work, and some means of flattening glued up boards wider than your jointer (if you have one). This is ultimately more satisfying, and doing so on a simple project like this would be very educational. I bet it doesn't seem so simple, though. Also there's the fact that if you build it of solid wood it will outlast you and your heirs. If of plywood, it will eventually be discarded. However, following such advice is often a prescription for getting nothing done, since the solid wood path requires numerous tools and other things.
Keep it simple. Because itís to be plywood and therefore by my definition impermanent, I would expend the least amount of effort on joinery. Instead rely on pocket screws and small nails extensively.
Tool stuff: Among the many other things you will need, get a router if you don't already have one (get three!). I hate the router. It's loud, messy, dusty, and potentially dangerous. It's also the most versatile power tool I know of. Every night take your router to bed with you (unplug it first) and recite the following: "This is my router. There are many like it but this one is mine. Without my router I am nothing; without me my router is nothing......"
I supervise a number of workmen and almost without exception they will cast about for any method of doing something other than set up a router, regardless of how much extra time and sloppiness is involved in the non-router approach. Usually after a while (years), some of them will begin to "think router", which often requires constructing a jig (another thing that will bog many a workman down). This is the point at which a carpenter becomes something more. I myself, even today, still try to think of ways to get around using a router, though I know better. Common sense must prevail though. I've seen guys get to the point where if they can't do it with a power tool, it can't be done. Like constructing some sort of sliding router sled to flatten a workbench. To my mind, that's over the top. [remainder of sermon deleted].
Design: If you wish a longer bench, consider adding perhaps two or three of the desired six or so inches to the case and the remaining three or four on the lid. It looks like the aspect ratio of the frame opening will get too long for a six-inch (17%) stretch. It's a Shaker looking piece and the overhang, elongated on the sides only, will not look awry. Not a big deal; your call.
Oak plywood will require to be edged at the top (at least three sides), and at the segmental cutout of the legs. Thus you must decide how to edgeband, solid vs. tape or a combination of the two. You certainly don't want to try solid edgebanding the curve at this stage. Oak plywood is made either rotary or sliced. Sliced looks better, is harder to find, and is more expensive. You should also consider that the legs of the piece will show both sides of the plywood, thus you should put the lesser grade face inward.
If you make the legs of solid wood, on the other hand, this cutout is relatively easy with your router mounted on a plywood trammel. But if you make the legs solid, then at least the bottom should be so as well, to ensure similar movement (cross grain construction), and down rolls the snowball.
For the face frame, you should simply purchase the flattest bestest looking 1x stock someplace and use it. BTW the phrase 4/4 typically, in my mind, refers to rough lumber, whereas 1x refers to ĺĒ surfaced material.
The back of the cabinet appears to be some kind of beaded material. This is available as a sheet product, e.g. plytanium, but you might not easily obtain it in oak. Iím not aware of any place you can buy oak T&G boards, though this would be preferred for looks. If you have to put some kind of oak on the back, you could use regular oak plywood, but itíd be better in ĹĒ than ĺĒ, and thatís more money. If you want to make you own T&G boards, itís not hard on the table saw, and looks shaweet. With a router table, you could make them edge beaded T&G, which really looks good.
At the size contemplated, the top will be subject to flexure in the center unless additional structure is added beneath the top. Fortunately this is easy to do. For this span you should plan to put two cross braces under the top, oriented front to back, and attached with pocket screws and glue. These braces will attach to the rear of the upper face frame rail in front, and to a lengthwise cleat in the back, about 2 ĹĒ wide, that runs the length of the case and is in turn pocket screwed to the sides. This cleat and the braces will be invisible when itís completed. Looking at the picture, the upper rail appears to be about 2 ĹĒ to 3Ē wide, and you should keep it that size. The front to back support braces should be ľĒ narrower than the front frame rail.
Joinery and Fabrication:
The face frame is most easily made by cross cutting to length and pocket screwing and glueing. The face frame clamp is a wonderful tool for this. Get the Universal brand, not the ShopFox. The face frame should be attached to the plywood carcase with glue and very small nails.
You could simply pocket screw the bottom to the sides from underneath. Itíd be strong enough, especially considering that the bottom frame rail and the back will lend substantial support. To grow your chest hair, dado the bottom into the sides. It sure makes assembly easier. To attach the top to the carcase, again Iíd pocket screw it. Do this upward from the inside face of the side pieces, and in two or three places on the inside face of the upper frame rail and the rear face of the back cleat. Glued and screwed it would be plenty sturdy.
From the picture, Iíd say that the curve on the bottom rail was inexpertly done. The best way to do this is with a router mounted to a trammel. Other ways exist. The segmental cutouts in the legs can and should be similarly done, with most of the waste removed with a jigsaw. I guess Iíd try to find a way to edge band those curves on the plywood legs. A regular clothes iron wonít do. Maybe you could use the non-preglued type of edgebanding, glue it and clamp with an inner tubes or rubber bands, or a shaped caul. When using a router, donít forget to set the distance to the far cutting edge of the bit, not the center of it.
Iíd recommend a rabbet in the back edge of the sides to accept the backboard(s). If you do this, rip the back edge of the bottom shelf narrower by the width of the rabbet, which equals the thickness of the back. This way the back board(s) will run down behind (at least to the underside of) the shelf, and can be attached from behind with nails or screws.
Presand all plywood before assembly. Watch for sand throughs; the veneers are thin. I expect youíll be using some type of varnish for durability. Use a sanding sealer, and sand it thoroughly without going through it. Maybe do that twice. Oak looks like the surface of Mars without this step. Make sure youíve no excess glue anywhere which could mess up your stain job. I personally donít like wiping varnishes. They never look right. Brush is the way. Linseed oil or oil/varnish mixes ainít bad either, and is easily renewed.