MIG would be easiest I'd have thought. If you're doing it all day every day you'll want a more professional machine. Just occasional use one of the "hobby" type machines will probably do. I've got a Clarke which is OK. "Gasless" wire is handy if you need to weld outside.
If you're using gas, the small disposable cylinders of gas don't last long if you're using it much. A refillable bottle of some kind is the answer.
You'll probably want an automatically darkening welding mask as well.
You don't say what sort of mesh it is? Weld mesh? Why do you need to weld it directly together? It's usual (in my admittedly not vast experience from some years ago) to frame "sheets" of it with angle iron or box section and put strengthening bars across the frames as well if they're much more than a couple of feet wide, as it gets pretty sort of floppy. Depends what it's for of course. So you could weld the mesh to the strengthening with short runs or "tacks". If you're making smaller box guards or something then bend it where possible and again use frames on the ends, even if it's only folded angle. gives you something to weld to.
Obvious really, but I wouldn't weld the wires together at right angles, ie horizontal to vertical. Be like knitting fog I'd have thought.
I'd weld two vertical or horizontal wires together side by side using short runs of weld as often as needed.
As always how you finally decide to do it will be down to you.
Final thought. If you're doing anything apart from the occasional bit of welding on uncoated mild steel, also if you're doing any welding at all on stainless and galvanised to name just a couple of the other common materials that need special precautions, you need to look at getting the right ventilation, extraction and/or breathing gear. You'll have to check on the exact requirements. There's something about it here
http://www.hse.gov.uk/welding/fume-extraction-rpe.htm