Opinions vary and I am not a professional welder, just someone that welds often. This is just my opinions based on welding minibike-sized mild steel, which is what you are asking about.
Arc welding (stick welding) is great for mild steel such as a minibike frame. It takes some practice to get good at it, mainly because the rod is long when you start a weld, then burns shorter and shorter as you lay the bead. Its a messy process, leaving behind slag and it may litter the area around the weld with bits of metal. Arc welding can easily penetrate all the way through to the other side of a metal joint. It can penetrate so good you blow a hole in the metal. It can also be used on less-than-perfectly clean metal, so its great for repairs and mods. 6011 rods will weld almost anything mild steel, even somewhat rusted metal, and in any position. 7018 rods make great filler on top of a 6011 weld, and can lay down a real pretty bead.
The flux in arc welding is on the outside of the rod. Flux is what cleans the weld metal, and shields the weld from oxygen. Burnt flux forms the ugly slag that must be removed after welding. Spatter is worst with arc welding, but cleans right off with a grinder and wire brush. In my shop arc welding is King, I can squeeze a rod between narrow areas, and can crank up the Amps for any amount of penetration. The main machine adjustment is Amps; your hands and eyes adjust speed, depth, angle, and width. For what you want to weld, and for the cost, pick up a used Miller or Lincoln 225 AC and DC arc welder. Its just my opinion, and my #1 choice for your budget and possible skill set.
MIG welding (wire feeder) requires a bit less skill once the machine is set up. Its cleaner than arc welding. Its perfect for production since the welding torch remains the same distance from the work, and virtually limitless wire can be fed into the weld. Pull the trigger and weld. Low cost MIG: flux-core MIG welders use hollow wire with flux inside. HF sells these a lot, but I dislike them. A "normal" MIG welding machine uses gas from a bottle into the torch to shield the weld. Spatter is less than arc welding. For thin metal work, like automotive sheet metal and exhaust tubing, MIG is hard to beat. Its great on minibike tubing and lays in plenty of filler metal in a hurry. Machine adjustments include welding power, wire feed speed, and gas flow.
Small gasless MIG welders, especially the 110 Volt units, lack the power to penetrate thicker metal. For minibike work, some folks say they are OK and I have noticed OldMiniBikes posts to that effect. A 220 Volt gas bottle MIG will totally handle any weld on a minibike, including repairs and mods to less-than-perfectly new metal. This fails your "simplest, cheapest way" test because a decent MIG welder and a bottle of gas costs more than a few arc rods and a buzzbox. Because the torch is big on the end, there are many places it won't fit, that an arc rod slides between.
Brazing is easy to do, because its more like soldering. An oxygen/acetylene welding torch needs no electricity. Heat the metal, get it glowing cherry red, flow in some brazing rod, and smooth it around. It is LOTS more forgiving than any kind of electric welding. The downside here is maybe strength, though I braze stuff all the time and its stronger than the parent metal. Bonus, you can use the same gases with a cutting torch, and slice metal any way you want. Bonus, you can use a torch to heat large areas of metal for bending. Brazing would be my #2 choice for adding tabs to a minibike frame. Gas costs a lot. A Henrob or Cobra torch and gases are amazing for minibike-sized stuff, but the price is up there with a MIG or arc welder.
Gas welding is a bit different, and requires a lot more skill than brazing. The torch is adjusted hotter, and into a smaller area, and a welding rod is dipped and moved along. A proper weld bead is left behind. You need real clean metal to do gas welding, and a ton of practice. Aluminum can be welded this way, I was taught by an old aircraft builder.
TIG is kind of like gas welding with an electric torch. It uses shielding gas and is a difficult technique to master. Farm it out. Want a beautiful weld, a "show weld" that you will be proud to exhibit, go TIG. Originally known as Heliarc.
Skill: Jody at weldingtipsandtricks.com is the man, he is all over YouTube. His welding videos are succinct and humorous, aimed at the average person who is a non-welder.
Cheap and simple: an AC and DC "buzzbox" welder like a Miller Thunderbolt, which is infinitely adjustable. Runs on 220 Volts and will weld anything on a minibike, has penetration, rods are cheap, welds are strong, and a little practice goes a long way. These things almost never break, so a used one is fine.
Simple, maybe cheap if you look, is a welding torch and brazing rods.
Jon