Well two things here, the gauge of wire goes with the breaker...doesn't matter what you're plugging into it. You could ofcourse use a lower breaker on heavier wire, just to be more safe, but nobody does that.
If it's a normal 110-volt welder then it will run on a normal house-hold outlet. However, most welders can pull up to 20 amps, and if you look closely house outlets are not rated for that. They are rated for 15 amps out of the outlet and 20 amps pass-through. The pass-through means the screws on the side and such can handle 20 amps going through them but that actual outlet can't handle 20 amps out it's own little holes. The only outlets that are rated for 20amps out their own holes will have a odd hole that can either accept a normal plug or one with one of the blades turned 90 degrees. The reason for that is if a device actually draws 20 amps, then it's sposeto have that turned plug so you can't plug it into a normal outlet, it would only be able to be plugged into a '20-amp' outlet. (I think welders don't have this turned plug because they don't always pull 20 amps...usually they draw less, so they don't have to have the wierd plug which would be a pain to deal with)
For houses it's normal to have a 20-amp circuit feed outlets and and 15-amp circuit to feed lights (some have 20-amps on both, some have 15-amps on both, depends on how cheap your home builder was). For a 20-amp circuit you have to use a minimum of 12-gauge solid copper wire. For a 15-amp circuit you have to use a minimum of 14-gauge solid copper wire. I say 'minimum' because you can go thicker, infact lots of people do, if you are running the wire a long way, might want to go thicker. The 12/14 gauge wire numbers are typically good to 100 feet from the breaker panel, so if you are going to be further than that away (remember extention cords also), probably should use 10-gauge solid wire, which is harder to deal with, more expensive, etc. If you've never done any wiring before I would suggest some books or just reading online, very simple, just work slow and question yourself if this is really correct. There are ofcourse rules and codes you have to follow but for what you're doing that's very easy to understand, if you are running wires in a conduit then you can use the THHN (I think that's them), the single wires, they take up less space in the conduit and so on. However to use those they have to be in a conduit from the outlet to the breaker box, no open accessable areas. Normally you'd use romex style wire, 2 or 3 conductor wires with a bare ground all in a sheath. You probably should know something about phases also, since if you are running garage stuff you can get away with less wires if you have to circuits on either side of the incoming 220volt line...stuff like that, or if you are wiring a 220volt outlet (which is actually 230 volt, and 110 is actually 115-118 normally, but nobody cares)
As for tripping a breaker, the best you can do (safely) is to just make your garage outlets the only thing on a new 20-amp breaker, so when you are using those outlets you know that nothing else is pulling power on that circuit. You can still trip a 20-amp breaker with a 110-volt welder though, I do it all the time with mine, on a dedicated 20-amp circuit, but that's the fun of a 110-volt welder
They do have such things as a 30-amp 110-volt circuit, but that is not a 'general purpose' outlet, which means you would have to have a breaker for that single outlet (only one, not even a pair), and I'm not sure if you can even get them with a normal plug, I don't think so, they are the round 4-prong things and all that. Also that would require 10-gauge wire, etc...
You should also put in GFCI (Ground fault circuit interrupt) outlets in, in any place where they have electrical codes they are mandated in any place near water or work area, etc. I have 20-amp (side plug compatible) GFCI outlets that I just put in, even in insturial gray

Those are very good outlets, and can and do save lives, and will not hamper any electrical device at all, no reason at all to not put them in, unless you want a house that won't pass inspection or like to have something dangerous...
Hopefully your breaker box has a master breaker that shuts everything down, not all houses do, my dad's doesn't have any main disconnect (except the actual power meter ofcourse, but you can't pull that yourself), so running new breakers is something to be done VERY carefully...trick is to put the wire on the breaker before it goes into the panel, then you can just pop it in and then turn it on...instead of fishing around with a screwdriver in a live panel...
If you want any more info, feel free to PM me, I'm not a licensed electrican or anything, but have done enough electrical work to go crazy...
Good Luck,
Mark Brodis