I did a 3 week intensive welding course in my teens; MMA, MIG, TIG, Oxy Acc, but I'm not a coded welder. I've done plenty of light welding over the years, so this is from a DIY perspective.
Upsides of MIG are point-n-click ease of use, very easy to create a good weld, non-live torch, lower light intensity. Post processing is minimal, simply release the welding trigger, allow to cool and put the paint on.
Downsides of MIG is it it limited to about 6mm thicknesses and can't be used in windy conditions, it has more moving parts in the machine and there's the cost of gas. Also the material to be welded needs to be free of corrosion and paint etc. or it will spatter like hell and can easy burn you.
Upsides of MMA are you can weld much thicker material and get better weld penetration, it is less sensitive to dirty materials than MIG and can be used for longer without overheating.
At a stretch, they can also be used for cutting, which is useful for confined spaces where an angle grinder won't fit - but the results won't be pretty.
They are about half the price of MIG units.
Downsides of MMA are many for the DIYer. It needs more skill and a lot more dexterity to use than MIG (a hell of a lot more if you're welding 3mm sheet or thinner, and even more if your elec supply is not up to it). Post processing is dirty and time consuming, and can be annoying - If you stop a weld run and then want to lay more on top, you have to stop and clear off the slag. Also the rods are always live, so you have to be very careful where you put the torch down. The light radiation is also a lot more intense than MIG, something to consider if your neighbours are close.
On that point - Both rigs will dim the lights in your house and maybe those of your neighbours, and they can intefere with TV and radio.
Do you have pictures / links re the forge?
If the forge requires you to weld together components of thicknesses greater than 5mm thick, this is not a DIY job for any part of the assembly which is safety critical. Sure, you could get them to join together but a weak join is easy to produce by mistake, and the forge will be subject to a lot of heat cycling, which can induce cracking.
£200 for a 'coded welder' sounds cheap. Last job I had done was an indoor BBQ in 10mm cold rolled steel. Basically a box that goes in a chimney and also acts as a lintel. Total runs of weld added up to about maybe 2 meters max, and this cost £350. My MIG might have joined them together, but I knew it wouldn't have the balls to give good weld penetration (oo err). The price also included a test piece which was cut and etched to test penetration - point being to test that welder, using that rig and that material on that day.
Back to the choice - Ideally a blacksmith would definitely want both welders. But, you can't go wrong starting with a MIG. The clarke welders shown in machine mart are ideal. If you go for one of these, buy the gasless MIG and get the conversion components to conventional MIG using 0.8mm (sometimes 0.9mm, for the DIY market) wire. Reason: the clarkes come with 0.6mm wire, which is for car body sheet steel. 0.8 will do more. Gas MIG is better, but Gasless can be used outdoors without fear of wind affecting the gas shroud, so it's useful to have both. The clark gas machines can't be converted to gasless, but the gasless can be converted.
Don't go for any that can't handle thicker wire, cos they won't have enough amperage. You will need at least 110A for armour type clothing, but you'll soon outgrow that. 150A would be better. Also important is fine control options. I guesstimate you'd need at least 200A MIG to weld a forge together.
http://www.machinemart.co.uk/shop/produ ... ig-welders