What kind of groove?
Dados: a dado grove is easily cut with a router plane. These tools are perfectly designed to do this job, and to give you a flat bottom with no tearout--something a tablesaw dado blade or a power router can not always do!. The secret is to cut both shoulders first with a crosscut backsaw. Clamp wooden guide fences to get a straight cut. Then define the two end edges of the dado with some chisel cuts. The router plane will incrementally clean this out in short order. If the dado is a bit thin--trim it with a side rabbet plane. If is is too wide---measure better next time!
You can also use a dedicated dado plane. The wooden ones can be found in almost any width for under $50.
Rabbets: rabbet planes are just about the most common plane available. You can get those for under $25 easy. With and without nickers. Moving fillesters (rabbets with nickers and a moveable fence and a depth stop) in the wooden form can be gotten for $50 or so.
The Harbor Freight plane will probably be a piece of trash. I have seen these. Much better to go onto Ebay and get a Stanley 78 for not quite as cheap, but still an excellent deal. The Stanleys have a lever depth adjustment of the blade. The Record 778 (if you can find it) has a screw adjust. These metal rabbets (or metal moving fillesters) do a tremendous job of cutting rabbets with and across the grain.
So it is clear that there are a multitude of great options for cutting grooves (did I say nothing about tongue and groove or plough planes??!!). And the great advantage is that you can learn a new skill (one that practically no one else has), plus hear yourself think!
