I would personally recommend MIG welding for ease of use, as it's not really much harder than using a glue gun! That being said, the MIG welder I use is a huge industrial grade one that we use to weld agricultural machinery back together with, so I couldn't comment on the quality of cheap ones.

I've used a stick welder before, and it was pretty easy to use and I'm sure wasn't very expensive (though having not bought it myself I couldn't say for sure).

It all depends what you can find; our big MIG welder was bought second hand relatively cheaply from an old factory that was closing down, and after a few quick repairs has been very reliable .. Worth keeping an eye out on ebay and the like to see what comes up cheap.