I have a welder/generator. Kohler Vtwin 10K generator and 230 amps arc welding. Way more than what a hobbyist in their garage would want. In the shop, I strictly use my MIG machines. Much cleaner welds and easier in general.
As far as a generator, I can't recall ever trying to run a mig off a generator, but I would steer clear of Harbor Freight. 95% of the stuff down there is junk. You would need a generator of decent size, like probably 7K or more I would think, and running a 220v mig, not a 110. I do know, my plasma cutter does not like generator power, causes surges and won't make a steady flame.
If you are dead set on running a generator, you might want to consider the Hobart 4500. It's a much smaller version of the champ 10000 and much more affordable. BUT, I personally would figure out your power situation first. Getting a MIG 220v machine is really the way to go. Monkeys can wire weld once the machine is adjusted and set for the work. Stick welding takes some practice to be able to do anything, and a lot of practice to weld difficult stuff, upside down, etc.
http://www.hobartwelders.com/products/generators/champion145/