You mean in a single pass right?
While a 220V machine can handle 1/4 inch easily, proper technique with even a cheap o MIG can technically handle the occasional 1/4" weld. I make the point because too many new weldors focus on max capacity even though they rarely use it. Kinda like HP with tractors. It just takes longer with smaller equipment.
I started to hobby weld a couple of years ago. So far oxyfuel and tig are my favorites because of the control they offer. Stick is the most challenging. Mig/fluxcore is technically the easiest to get started with but to this day I have trouble really seeing the puddle through the smoke. Mig/gas solves that but is more $ and limited field use.
My 2 cents: take an oxyfuel course first because it teaches the fundamentals and is slower and more controllable for a newbie. The other techniques have many advantages in practice (sometimes) but oxyfuel is the best way to start.
Good advice....The thing that strikes me most about the OP's opening post was in the first 3 or 4 words, the desire to
learn. There has been a lot of good advice passed along in this thread but it seems to me a lot of it is somewhat misdirected....This is my opinion only of-course and is not meant in a off handed way it is just an observation on my part.
If someone truly wants to learn a new trick you have to start out walking not running full blast. Where it is possible to learn to weld by just picking up the stinger and making contact with the medal you are working with but you are putting the cart in front of the horse and I guess you could train a horse could push a cart down the road but that would be mainly for show. If the OP truly wants to learn the art of welding he must first get the basics down pat and then take this information home with him and burn rods and or wire until you get the hang of it.
It really doesn't matter what process you use because any of the processes can handle 1/4 welds with ease with good equipment. The reason I would not opt out for a wire feed machine as the first choice is that they are easier to get results from and the user would tend to get lazy and not get the full picture of how the process works.
You could weld just about needed for around the house use with a good AC/DC stick welding machine and an acetylene torch setup. In fact you could just skip the welder altogether and gas weld the parts for most uses. True this takes time and in most cases is not cost effective but it would be nice to know for the time when the nice new inverter machine has blew out the control board and you have to make an emergency repair on something you need. It's always a good idea to be able to fall back on the basics in times of need than to try to learn them in an emergency situation.
With the budget the OP has stated he wants to stay in he will almost have to stay with the basics, $500.00 won't get you far when you are talking about wire feed machines that are able to make consistent welds on 1/4 inch or heavier material. You can spend $500.00 on safety gear and consumables alone, then you have all the other tools that are needed to do most common jobs and that's just the start. I don't want to make you shy away from wanting to learn to weld, I'm just pointing out it that welding can be an expensive hobby just like hunting and fishing or anything else you do to pass the time.
I can't stress enough the need for the proper safety equipment because just one bout with your, or a family members eyes being burned and the pain it causes is a bad way to learn. Not to mention skin cancer and breathing problems that can be added to the mix. You will learn about all these things and more and how to prevent them from a well trained staff instructor at a local tech center.
I know these things because I am a self taught welding amateur and have spent the nights awake with potato wedges taped to my eyes trying to find relief. Then several surgeries removing the bumps from my eyes that welding with cheap welding gear can bring. Then you add in the many trips back and forth from the doctors having skin cancers removed from my arms, legs, face, neck, and ears from not wearing the proper gear. I learned to be a pretty good welder but it came at a price and the sad thing is that it all could have been prevented. It's like cigarets you don't know what they are doing to you until it's too late.

Well you do but these dangers need to be drilled into your mind so that you can teach your kids the wonderful art of welding the proper way.