I can't say that I know much about welding but this is a topic that comes up over and over on diffrent bicycle forums. One thing I have learned from reading these discussions is that a wire fed MIG seems the best way to go for bikes. The general concensus on TIG is that it is too powerful for the thin walled mild steel that our bikes are made out of...Kelly
Not at all. Actually I have been a machinist and shop welding for many years and I can tell you what actually works best.
The 120V MIG's are the best overall purchase for the average guy for ease of use and a short learning curve. They require no improvements to your electrical load panel and can be flux core or solid core wire feed, depending on the make and model you buy and how much you want to spend.
The drawbacks are that 120V welders tend to be weak on the amount of penetration they provide, depending on the manufacturer, which is why you'll hear some people praise them and some people can't get good results and think they are terrible at welding when in fact it is just because the welder is not adequate to what you're working on.
With that, I am not expert on whose product is the very best, but I have never heard complaints about Miller or Hobart products.
For the types of work most of the people here would be using them for, you actually have several excellent options depending on your skill level and the thickness of your wallet.
TIG is actually the very best welding process for anything other than production because it allows very fine control over the heat (torch) and the amount of filler material (rod). You can weld very thin material with TIG because you can shape the electrode tip to a fine point like a pencil and control the heat so low that you can lay a bead that is merely 1/16th inch across or less (yes, I've seen this done with some good equipment).
I have owned a couple TIG welders and have had excellent results with them, however you have to learn how to use it or it is no use to you. It also offers more variety of welding non-ferrous metals, such as aluminum, copper, brass, etc. so long as you can obtain the proper filler material.
Stick welding is okay for general purpose welding but you do not have as much control over the heat of the weld arc and it is extremely difficult to weld thin material with. Additionally, you have to deal with slag caused by the shielding chemicals on the rod that must be chipped away and cleaned after you make a weld. Stick welders on not recommended on thin materials.
Probably the most overlooked and cheapest welding is gas welding. It is as cheap as a small wire feed MIG welder and the process is similar to that of TIG welding except that you are using a gas-flame instead of an electric arc.
You do have control of the heat, like TIG but not on the fly. You actually have to get acquainted with the flame cone and know which areas are hotter than the other and to get a feel for where to place the tip. But you still just add filler material to the work as you heat up a small molten pool of metal. You can even use uncoated wire coat-hanger to weld with. You may also use brazing rod and a brazing tip on your torch to weld parts together. It is not quite as strong, and technically not a weld but many bicycle frames are brazed together and hold up quite nicely.
For anyone doing this, I would recommend learning to solder copper fittings together first so they understand how to heat metal and get a feel for how solder will flow into a joint using flux, for this is much the same as how one would braze metal together.
Hope this helps.