Believe it or not technique for using an arc welder welder to cut metal or pierce holes in metal is covered in most Lincoln AC-225 buzzbox manuals or Miller Thunderbolt buzzbox manuals.
In short, they recommend kind of a sawing motion with the electrode after punching through. Diagrams and techniques are in the manuals available for free download.
6010 (DC only) or 6011 (AC or DC) are the best rods to use for improvised cutters because they penetrate deep, and deposit very little metal in the process. A rod like 6013 or 7014 would not be the best to use, because they will deposit more metal and are not deep penetrators. Of course any rod will burn through if you crank it up high enough and the metal is thin enough.
Generally speaking you are not going to want a 1/4" diameter electrode unless you have a really powerful welder. You will usually run the rod at 2 to 3 times the setting you would weld with so with only 200 to 250 amps I am thinking 1/8", 5/32", 3/16" would be a real stretch and about the limit for practical cutting with decent results with the amp supply at your disposal.
With practice it is possible to get semi-decent with cutting and piercing holes. Cuts will not be plasma quality, but can often be good enough for a backyard farm shop. It is relatively cheap to use as a cutter - no doubt.
There are also, special cutting electrodes for small arc welders offered, but they can be pricey (I have never tried any of these special electrodes due to high cost. I save my old damp slightly rusty 6011's for cutting duty).
Gouging with something like an ArcAir gouge takes a really big industrial welder and a really big air supply and is not precise tool at all. Very noisy and extremely messy too. Primarily only useful for cutting out bad welds in thick metal.
Of course if the hole needs to be precise then lots of good recommendations have already been made. Another alternative option is annular cutters. These work sweet but must by used in either a mag drill, drill press, or mill. I do not own any of these but have used them on occasion at work - much better than a holesaw but cost is more too - but still reasonable as they will last much longer than a holesaw. Annular cutters have been covered on TBN before so useful threads available with a search.