If you're going to weld more than a go-kart frame, I'd recommend you comb ebay, craig's list and your local want-ads for a used mig welder. You can get more welder for your dollars that way and there is little bad to say about Lincoln, Miller, Clark and other "professional" brands.
Mig welding is a process where the welding "rod" is actually a wire fed through a gun-like device, adding material as you weld. "Regular" mig requires a shielding gas such as argon, Co2, or "tri-mix" which is a combination of three gases. Flux-core wire welding uses the same machine, wire fed through a gun, but the wire has a flux that when heated by the weld, makes a small cloud around the weld puddle and shields it from the atmosphere. The machines are cheaper, the materials are cheaper because you don't need to fill large bottles with gas, but is a bit messier in the weld area. Expect to do a lot of cleanup with your grinder.
Stick welding is a cheap way to get started with welding, and basically the unit is a giant transformer with two thick wires - one with a ground clamp one with a "wire clamp", which holds the welding rod. Stick welding has a lot of advantages, the primary one being you can weld in any kind of weather, and the material you're welding doesn't have to be completely sanitary. Obviously with all welding processes you should clean what is to be welded but stick welding works fine with rusty bits. Been there, done that.
There's also gas welding, which requires a welding torch and two bottles - one full of oxygen and one full of acetelyne. The two gasses combined (while burning) produces an incredibly hot, reasonably precise flame and you can weld any kind of metal with the same setup. You simply use different welding rods for the different materials. It's another great process that you can use anywhere as it's ineffected by environment and you don't even need electricity. A lot of people who weld "in the field" prefer this method because all you need is the gas, the torch, and a stryker to ignite the torch. You can also purchase different sized torches, tips, cutting heads and so on. I've seen jewelers torches that could fit in the palm of your hand for the really small stuff to the usual 3' long monsters that weld teeth on backhoe buckets, the backhole bucket being the size of a pickup truck and then some.
A decent victor knock-off torch (victor is a name brand that's very good) will run you about $200 and a real victor will run you about double that new. It's a cheap way to get into welding and you're not limited by thickness, material type, etc. But you do have to buy gas often if you weld a lot.
Like everything else in life, it's a bunch of tradeoffs. I prefer fluxcore wire welding because I dislike the hassle of refilling tanks as mine always empty out during late night welding and I can't refill them at 2am. It's also a one-handed process unlike most other styles of welding so with my very shaky hands I can use both hands on the gun and actually have some control. Most people do not have this issue.
The absolute cheapest way to weld is to buy a pair of car batteries and hack off some battery cables from the various junk cars. Put the batteries in series with one cable, and use a second cable between negative and your work/project, and the positive post goes to a third wire that you attach a big springy clamp to. Put your stick-welding wire in that clamp, and have at it. You control the weld puddle by insertion - meaning the faster you jam the welding rod into the puddle, the cooler the weld. The less you shove the rod in, the hotter the arc will be.
This is how *I* learned to weld. It's fun actually. A friend of mine to this day welds using a 120A alternator driven by a vertical 18HP lawnmower engine with three belts between the two. All free stuff he scrounged from who knows where. It works very well to about 3/16" thick materials without any issue.