For the tongue-and-groove joint, there is only one hand-tool method that I teach, and it takes a specialty plane: the No. 48 (for 3/4-in.-thick or larger stock) or No. 49 (for 1/2-in.-thick stock). Both planes cut a 5/16-in.-deep groove (and matching tongue); the plane bottoms out, and stops cutting. I never put this plane – also known as a “match plane” – on my tool lists, though, because I don’t want to scare students with such a specialized plane. I just loan mine (and typically at least one or two students will have these and loan them out, too).
But in case you do want one: Old Stanleys are nice because you can adjust the blades independently to cut the interior tongue shoulder a hair deeper than the outside tongue shoulder, to pretty much guarantee a tight fit on the joint. New Lie-Nielsens are nice because the one-piece blade means it’s a little less trouble to sharpen (just don’t cinch down too tight on your sharpening jig, if you use one, because it’s possible to squeeze the blades out of square to one another). There are other metallic match planes, such as the Stanley No. 148 (which looks to me like Dr. Doolittle’s Pushmi-Pullyu), the Union No. 41 (which became the Stanley No. 48 after Stanley bought Union) and others, as well as wooden versions, both vintage and new.
One thing that is important: Lay out your backboards so that you’ll be planing with the grain and mark them (I use a cabinetmaker’s triangle). Then fence off the same side of each backboard as you cut the joint (the marriage mark helps you get it right). That will ensure the joints will go together, even if (when) the plane isn’t cutting dead center on your stock. And if the boards are running horizontally, arrange things so the tongues face up and the grooves face down – theoretically, this makes the joint more dustproof and waterproof. If running the backboards vertically, I don’t think it matters in which direction the tongues/grooves face.
And just as with the skew rabbet plane when cutting shiplap joints (among others), start at the far end and work your way back as you bring the cuts to full depth. This will help you stay on track. And don’t grab the knob; the base of it is good for hooking your off-hand around as you push in and forward. If you grab it, you’ll likely tip the plane. Don’t grasp the handle tightly, either.
p.s. I know Lie-Nielsen Toolworks is out of stock right now on both sizes of this plane. Once it’s safe to have a full staff on site again, I’m sure that will change.