Looking at this again, I can't help but think OffRodeo is renting this new place.
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I will be moving into a house with a garage

So I can't stress this enough..
GET PERMISSION FROM THE LANDLORD!
Or you can end up in legal trouble!
A rental unit MUST have such modifications done by a licensed/insured/permitted individual or the landlord will end up liable for loss of life and/or property, I am pretty sure he wouldn't take that chance. And his insurance company would remind him if something did happen.

This is horrible advice:
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I would still run the cord from the dryer if it were me, at least for starters, it can be very annoying but at least you get started. Building a garage is just like building a vehicle if you don't do it piece meal then it doesnt happen. I started doing the dryer cord, after which we ran the outlet to the garage. Would I run from the dryer again?? Yep! Was it annoying to run back and forth to the breaker ?? Yep, but not nearly bad enough to stop welding.

This behavior is why so many house fires are blamed on electrical.... uh, I don't know what happened sir, the wall just spontaneously combusted...... <img src="/forums/images/graemlins/shiner.gif" alt="" /> <img src="/forums/images/graemlins/shiner.gif" alt="" />
I have personally witnessed the consequences of trying to run welders on a system not capable of the load and not properly grounded, they had a catastrophic meltdown. When individuals who think, hey no problem-I'll just hook this up here, without knowing the size of said circuit and without proper grounding, do so on their own, causing it to literally meltdown. Believe me, they got a proper @ss chewing and were back charged, not cheap.
But I digress.. if you are renting, talk to the landlord, he is the one that actually has to make the arrangements with a contractor. He might split costs with you, if he goes along with it.And I'm afraid that this will hold true:
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I think that you will find that nearly all landlords are not going to authorize you to put a 220 volt in, especially when they find out what you intend on doing with it. The threat of fire, damage to their house, visions of clunkers in the driveway, etc.

The other options mentioned, like the onboard might be better for you. Or one of those bad boy trailer mounted suckers. <img src="/forums/images/graemlins/mrt.gif" alt="" /> <img src="/forums/images/graemlins/mrt.gif" alt="" />