Cutting gauge, knife, chisels, saw and router plane can do grooves too but you will be happier with a plow plane of some kind.
Metalic plow & combination planes are plentiful and inexpensive (relatively) on the secondary market. Wooden ones (in good shape) are a bit more expensive and there are more things to check before purchasing than a metalic one so I’d stick with looking for a metal body plow plane as your first go.
The Record 044 and 044R are nice, especially if you find one with all of the blades as later model ones came with both Imperial and Metric blades (better match to chisel sizes). But those made by Clifton, Stanley, Union, Seigley, Mongomery Ward (actually Stanley castings) and a host of others are also fine.
Probably the most common non-bench, non-block plane on eBay is the Stanley 45 in some configuration. As a plow plane, you want one with its extra skate, at least a few blades (these are easy to find by themselves or even make), at least the long rods (2), a knob (either on the main stock or on the accessory skate, depending on the vintage), a functioning blade clamp, the main depth stop and a fence that is true.
Later models of the 45 have some nice features like a micro-adjustable fence, secondary depth stop, more adjustable blade mechanisms, a beading fence and a few others.
Often missing (but seldom missed) is the slitting cutter and its depth stop. I however have found it to be a clever addition. Just watch your knuckles when picking up the plane body. Frequently missing is the support cam and frankly, it isn’t missed as it is a pain in the butt to use. Also frequently missing are the knickers but replacements and replacement screws are available. You can live without them for crossgrain work if you just knife a line separately.
In short, just look for a $50 plane with a few plow blades (1/4″, 3/8″ and 1/2″ will get you most everything you could ever need) and spend a little time cleaning it up and experimenting. But if you are patient, I’ve seen several 45 “kits” go by at the $100 to $150 that are very nearly complete, clean and ready to work and I’ve seen 100% complete pristine kits go for much less than $200. Record and Stanley 44’s and 50’s seem to be a bit higher for whatever reason. But you can’t really go wrong with a 44, 044, 044R or even the 43 family either.
One other comment on the 45 combination plane design (also the 55 design but frankly, that thing is just too nutty to be as useful as the vintage advertising would have you believe), with good sharp blades and good stock selection, the beading blades of the 45 work well up to about the 3/8″ size. So, later when it comes time to add the bead to the clock you can make your own beading scraper, use a commercial scraper, a dedicated wooden body beading plane (these are generally the best for several reasons) or use a #45 or #50.
So in the short term, a #45 in good shape will carry you very far along as you research and learn more about the other options out there.