Welding transformers are specially designed so they don't blow a fuse when the secondary is shorted. You could use a standard transformer with an inductor in series with the output. The output voltage must be less than 50 volts to avoid getting fatally shocked. The required output current depends on how big a thing is to be welded. 100 amps will do most jobs altho commercial welders usually go higher.
actually the amp setting depends on the size rod you are using. A good rule of thumb is to set the amps to the decimal equivelent of the rod. so a 1/8" rod is .125" decimal, so you would want your welder at 125Amps. not that it's really important...
Also, the heat that fuses the metal is generated by the arc jumping through the air.
The mig welders require the gas supply. They use the same prinicple to do the welding, but the gas (Co2 or Argon) shields the weld from all the chemicals in the air that would bond to the weld. without the gas you get a weld that is all pitted and not very strong. and for the arc welders the shielding gas is created by the chemicals that are on the rod.
During my senior year of Highschool, I took a nightschool class taught by the Iron Workers union. I even got Ohio Bridgecode certified in it.
I have a small wire feed welder, a MIG or metal inert gas (iirc)
I don't use gas but flux core wire. The flux makes enough smoke around the arc (to shield it from the o2 in the air). It make more of a mess (spatter) but works in a breeze better than gas.
My machine only occasionaly trips the 15amp gfi breaker.
It has a wire speed knob and an amp knob. fooling with them gets the action right for the material.
The big stick arc machines rely on the flux outside the stick to make smoke to sheild the molten metal from the air.
There are dozens of kinds of sticks.
Some early rods were like coat hanger wraped in news paper.