I actually find TIG to be the easiest type of welding, though I've been known to be weird. I'm no expert, but I'll explain my experience with all 3 main types of welding:
I find MIG great for tacking things together, but it's often messy and fume-heavy unless everything is done perfectly. It works by melting/vaporizing the filler material and "shooting" it at the workpiece to join the host material. That's why it can sound sputtery sometimes. It's also an all-or-nothing kind of welding. There's an "on" button, and that's it. So if there's anything intricate that varies in thickness/etc..., you'd best be served by a TIG machine.
Stick welding doesn't really apply to anything gun related, so I won't go there, but suffice it to say that stick welding is akin to throwing huge amounts of molten metal and flux at a joint generating a huge amount of heat and smoke. It's best for large scale pipe/ship welding where the hardness of the parent material may not be as critical because of the larger heat affected zone generated by stick welding. (unless you use special rods that solidify into a hardened state). If the part is going to be heat-treated afterward, that is another story.
TIG, on the other hand, is 100% controllable and clean. Everything about it can be adjusted, and usually on the fly. Like others have stated, gun work is usually done at less than 150A (less than 75A most of the time), so that machine should be fine. I would use a 3/32" 2% Lanthanated tungsten, as it usually has the best combination of low burn-off, thermal cycling capacity, and pulse stability. It's also non-radioactive. You can sharpen it different ways to have different arc characteristics. A thinner, sharper point tends to penetrate less with a wider weld bead, while a slightly blunter tip will produce a slimmer and deeper bead. This pdf explains:
https://www.arc-zone.com/pdf/GrindingTungsten.pdf
Look up "type 17 tig torch stubby gas lens kit" and buy it - trust me it's better than the normal stuff.
A foot pedal or hand-piece wheel adjusts the baseline "heat" of the arc, while the hand holding the torch controls arc size/voltage/current, etc.. by varying the distance between the tungsten and the work. This level of control allows you to contain the heat to a very specific area or bathe a whole area with heat. Try to keep the tungsten as close to the work as possible with a small, safe margin to prevent dipping. You'll find it after a little practice. Even the best TIG guys will dip their tungsten from time to time though. Sometimes metal just flows in ways you don't expect.
The other hand holds the filler material, which should be added to the parent joint after they start to melt together and form a "keyhole." You'll know when you see it. My very favorite filler for general use is called ER312 Stainless. It is used to join any combination of steels with a strong joint. Usually, the joint ends up being stronger than the parent metal. A couple of pounds will last you a long time too. For gun stuff, 1/16" is a good filler size, but it's good to have 3/32" around too. I buy from McMaster Carr because it's easy, but you can probably find it cheaper if you look around. Both ER70S2 and ER70S6 produce fine welds on just about any steel (even stainless in a pinch...sshhhh don't tell anyone I said that). ER70S2 does contain more deoxidizers to allow a dirtier parent material to weld safely, but S6 does flow better as TRECustom said. You can feed the filler from the front of the bead or from behind based on different setups, but most seem to feed from the front.
If something has a silvery/zinc/chromium/yellow finish (or any coating at all), avoid welding it at all costs. If you MUST, wear a P100 respirator because the fumes are dangerous and will make you sick if you don't filter them out. Ask me how I know this...
In short, TIG welding is the "hardest" because there are so many things that must be balanced to do it right. But it's also the easiest to get great results with IMHO, because you have control over everything and you can work at your pace. MIG and Stick work at their pace (usually) and you have to keep up or the weld will suffer.
This website is basically a welding class for free - I highly suggest spending a few hours going through everything on it:
http://www.weldingtipsandtricks.com/
If you have any questions at all, I'm happy to lend whatever (limited) wisdom I have on the subject.