Hi Hooked, I'm a beginner as well - well I've been carving since august so not totally fresh. I've tried the following: mahogany, red alder, birch, and basswood (or linden, I think that's the same), and also cherry which is great but expensive. If I were to carve that head, I would probably use basswood for the actual head and birch for the antlers. Basswood might be too soft for them. Red alder is somewhere between birch and basswood, in my experience, but I suppose it tends to warp when drying. With mahogany, you'd want to avoid the sort with wild grain, or ribbon grain, I suppose it's the same thing. It looks great but the grain changes direction pretty much, so it's hard to get a smooth finish without sanding. (Someone correct me if I'm wrong at any of this, as I said I'm quite fresh.)
Also - with your apparent talent, why not try it as full sculpture, instead of flat like you have done now? I'm sure you would succeed at that. You'd have to make them smaller of course, to save material, but I'm sure the result would look great.
You gave me an idea: I'll make some 3-dimensional head like that, and use carved branches for antlers... won't be symmetrical but fun, I'm sure. In fact that's just what I need above our coat hangers - my wife messes the place up with scarves, she can hang them there!
