I use a Hobart Handler 140, which is about the biggest size you can get that runs on 120V. I started with an $80 Italian wirefeed welder that had only two power levels and no feed adjustment. It was a good little welder ... until I fired up the Hobart for the first time. What a difference!
A good quality welder is worth the extra money. Brand names are usually a better bet. A lot of cheapies use aluminum wire in the windings instead of copper. Duty cycle is worth checking in your search - just tells you how long you can weld vs cooling time in a 10 minute period. I've never had any trouble with overheating. I usually end up welding for less than a minute, then stopping to turn something around before welding again.
I'd consider stretching the budget, even if it means delaying your purchase for a month or two. That's easier than trying to sell an inadequate welder so you can get the one you should have in the first place. Keep in mind that fluxcore will get you going right away. Adding gas will cost about your whole budget right now. Definitely check the ads. There are a lot of good welders out there that have never been used much and are now showing up for sale.