Working from the materials list, begin by edge-gluing enough wood to make all the large panels you'll need, including the top, bottom, inside gables and shelves. If you have wide boards on hand, save time by making these parts from one piece. Otherwise, laminate the panels and let them dry overnight before surface preparation and cutting. Many of the newer thickness planers can accommodate even the widest of the parts required (e.g., the top), so wait until after glue-up to do your planing. Ensure you remove all glue squeeze-out from the wood before you mill it.
When it comes to making the columns, you have two options. You could glue up standard 3/4"-thick pine for these parts or mill thicker wood down to size. For the amount of wood you need for this project, consider using construction-grade lumber. A 2x6 is really 1 1/2" thick (just right for the 1 1/4" finished size you need), and with careful selection you can find some nice wood. Choose kiln-dried material and you'll be able to use it with minimal drying time.
Start from the top
Begin by cutting the top and bottom to finished dimensions. First, joint one edge true, then rip the wood to final width. Finish up by cutting the ends square. Take your time: clean, crisp surfaces are essential to an attractive project. The all-important bevels come next, and you have a couple of options for preparing them.
The first is to tilt your tablesaw blade to 45º, then cut the bevels required along the longest sides of the top and bottom pieces. Bevel widths differ for the top and bottom, as shown in the plans. Repeat the same operation on the ends of these parts, using either a tablesaw or sliding compound mitre saw. Both the top and bottom members are too large to handle accurately on a typical tablesaw's mitre gauge, so use something more substantial. Many workshoppers build a wooden cross-cutting sled for themselves, and this can do a terrific job.
You can also cut bevels with your router. A large and sharp bearing-equipped 45º chamfer bit does a good job on most wood species. Begin by routing one of the short ends first, taking a series of passes when creating the larger top bevel. Work your way around to the adjacent long edge, followed by the second short end. This order of events automatically removes any splintering that would occur as you round that first corner with your router, helping to create the crisp bevels that make the project look good.
Next come the four columns, which you can create from either thick stock or glued-up laminations. Mill them to final size, then sand them with up to 150-grit abrasive. Instead of bevels on these parts, ease each corner of the columns with sandpaper wrapped around a wood block. Your micro-bevel should be just 1/16".
Now cut the inner gables and shelves to the finished width and length. For best accuracy while preparing the corner cutouts in the shelves, place one of your columns on top, flush with the outside end and side of the gables, then trace the column and cut on the waste sides of the lines.
Get sanding
It's time to make some sawdust. Most planed lumber comes from the mill smooth, so unless your wood is really rough you can start your sanding campaign with a 120-grit abrasive, moving up to 150-grit. Regardless of your starting point, a random-orbit sander is best for this work.
If you're planning to stain your dish rack, apply only light pressure on the sander. Random-orbit sanders don't create a lot of cross-grain scratches, but they're not perfect. Wood that looks and feels smooth when bare will show unsightly swirl marks after staining if you apply too much sanding pressure. A light hand gives the best results. Just to be safe, use only a handheld sanding block to smooth the bevels themselves. It's too easy to ruin the bevels with a power sander.
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Where is the Illustration?