I built the vanities for our new bathroom with no plans, just a picture like you have there. They didn't have the the drawer at the bottom but to me that's just a minor detail. They were the first ones I have ever built and they came out just fine for me. Try it out of mdf first, if it comes out you'll know how to do it with plywood and real wood. Then paint the mdf one and sell it at a garage sale or something. Mdf is cheap so you won't be out much and you may get your money back.
I also do not use plans for a cabinet like this. I generally make the base separately and then make a box for the rest, connecting them with screws. One nice thing about doing this is you can level the base then attach the cabinet to it. The last bathroom cabinets I made I used melamine particle board. It is not the nicest thing to work with but the finished project is durable and the material is cheap. If you rabbett the sides you can just screw on a bottom and some ledgers across the top and the screws do not show. I also screw partitions through the top and bottom but to make a bottom drawer, you would need to clamp the horizontal false bottom piece in with glue and no fasteners if you use melamine. Otherwise a few brads would hold it while the glue sets.
I would know the size of the top first if you want to use one of the one piece fake marble tops. The top should overhang the cabinet 1/4 inch or a bit more. The height should be 32 to 36 inches depending on who will use it (I like 36 but if it's a kid's bath, you should probably go smaller). I size drawers in even multiples of the dovetail template I decide to use (even inches with the 1/2 inch template and 7/8 intervals for the 7/16 template). Dovetails are easy with the right jig and add a bit of class at very little cost. To use commercial slides, the drawers must be a 1 inch or a little more narrower than the openings in the cabinets.
Umm... I'm not sure about your house, but around here... my supply lines come out of the floor. That drawer wouldn't work for me without rerouting the lines into the wall.
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