My first forge was made from a brake drum and some other steel scraps. Here's how I did it, and how to avoid some of the mistakes I made.
When I first started trying to teach myself blacksmithing, I read all the books I could find. The Internet was just beginning to exist, HTML and browsers did not yet exist, and there was no Google or YouTube. So I read books by Alexander Weygers and articles in Firefox, do-it-yourself books from Readers Digest, and anything I could find.
All of them agreed that a working forge needed an air blast (which could be from the bottom or side) and a depression to form a nest for the fire: a fire pot. Several described using an old cast-iron brake drum from a truck or car to form that fire pot. I went to a salvage yard and bought a used brake drum cheap. I found some angle iron in a scrap pile, and I had some steel sheet (16 and 18 ga.) lying around from some armor projects. A nearby surplus store had a nice dual fan unit, also very cheap.
I put together a square angle iron frame by riveting each side to triangular sheet gussets, sized so that the brake drum would rest on its rim inside. I riveted angle iron legs to the frame, and used some bars to brace them to prevent wobbling. The sheet steel panel for the fan switches acted as a brace for its side. On the bottom of the brake drum I bolted a pipe flange, and used small plumbing joints, elbows, and nipples to connect it to both of the fans, with a pipe leading downwards to a screw-on cap to act as an ash dump. Near the top of the pipe, just below the brake drum, I put a round bar with a piece of flat sheet on it, to act as an air dam (like a butterfly valve). The frame was very sturdy, and most of this worked well, but there were two main problems:
The "ash dump" pipe cap rusted solid, and doesn't open. I should have made a hinged lid instead.
The air control was too sensitive. A tiny change in setting would result in a large change in air flow.
To protect the interior of the brake drum, I lined it with some refractory fire clay. Since none of the books had explained clearly enough what shape the fire pot needed, I put down about an inch thick layer. In the center, I capped the air hole with a small circular grating, made of a 1/2" thick round slab of steel with seven quarter-inch holes drilled through it. After I'd built a charcoal fire and let it burn for a few hours, the clay was as hard as I could ask for. The shape turned out to be difficult to use, though. The fire was never deep enough; I should have sloped the sides down smoothly from the edge to the center, and had the center grating resting on the bottom of the drum instead of an inch above it.
I had planned to make a small chimney out of #10 cans, and put a conical cap above the firepot to act as a smoke hood. The chimney never worked well, probably because there was too much air gap below it and too little draw above (I never made it more than about 5' long). The hood, though, was very convenient for working outside, as it gave the forge shade. If at all possible, don't forge in full sunlight, because it's just too hard to judge the color of the metal.
The fan was very effective, perhaps even too strong. I had put in a pushbutton and a 3-position toggle switch, so that I could have the fan off, controlled by the pushbutton, or just plain running all the time. As it turned out, the pushbutton wasn't convenient to use. If you're not using a hand-powered blower like a bellows or a cranked model, you'll want the switch to be something you can operate with your hands full. My current forge has a momentarily-on foot pedal, which I like better for an electric blower. I'm working on a double bellows.
There was one other significant problem with this forge design: no fuel storage. Small rivet forges have a wide flat area surrounding the fire pot, with a raised rim. Larger, less portable forges have an even larger wide flat area, usually with a raised edge. The flat area gives a place to store extra fuel ready to rake into the fire, and (if using coal instead of coke or charcoal) a place for the coal to turn into coke. Because of the shallow fire pot, and having no fuel storage space, I would continually have to throw a little bit of new charcoal on the fire. Every time I moved my stock, the charcoal would tumble off the firepot. It was very frustrating.
Would I recommend a brake drum forge for a beginner? Certainly. Mine was good enough to get me started, without spending a fortune. But there are mistakes that I would avoid, were I doing it all over again.