It's not too hard to do. Take the front panel off and pull a breaker straight back off the board. Once it's unclipped from the central bus bar and just hanging from it's black wire, it's dead, just loosen the terminal screw to get the wire off. Take the breaker to Home Depot and buy four 15 amp slimline breakers. They will look like skinny versions of what you just took out, and two of them will fit in the same space as a single regular breaker. There are some types/brands of breaker boxes and breakers that will not accept slimline breakers; if you can't find Square D or GE slim breakers that match yours then you are SOL. But most likely you will have no problem. Also buy a twin 220V 20amp breaker, and however much 12 gauge Romex cable you will need, and a box and cover and 220v female receptacle that matches the plugs on your compressor and welder.Also strain reliefs to protect your new cable from the edges of the box you are buying and also where your cable will exit the breaker box. You will need to replace four of the 15 amp breakers in your box with the slimline breakers, leaving a space that the double breaker will fit in. While you have the breakers out of the box, you will see that the central bus bars that the breakers clip to are made so that, as you clip breakers on, they draw power from alternating sides of the central bus bars. These sides of the bus bar carry 110 volts each. Your new 220v double breaker will clip in so that it draws off both of the center bars at once, and will ONLY clip in like this. Connect the black and white wires from your new cable to your new double breaker; pretend the white wire is black and hook it up to one terminal and connect the black to the other terminal.Install the bare copper ground wire from your new cable to the same bar that the other ground wires are connected to. In general, as you work, look at how the electrician installed things and copy it. The only thing you will not copy is that your new white wire will connect to your new double breaker instead of to the bar where the other white wires from the 110 v circuits are connected. Use the strain reliefs just like the electrician did, and staple the cable to the structure just like he did. mount your new receptacle and box wherever you want. You will now have a 220v 20 amp outlet, which will safely run either your compressor or welder, just not both at the same time.
Wiring to two single breakers like you thought of in your first post is dangerous for several reasons: there is no guarantee both will trip at the same time, and you would have to be sure the two breakers were getting power from different sides of the central bus bar.
All these directions assume that your current breaker box is full. If in fact there is room for more breakers but the knockouts in the front cover have not been broken out, just knock out two and install that 220v 20 amp double breaker. You won't need to replace existing breakers with slimlines if the box has room to take the double. Just take a breaker out and go to Home depot to buy a matching double.GE and Square D both make two kinds of breakers so it's easiest to just take one with you to make sure you get a matching breaker.
Of course, turn off the main breaker while you are running wires through the breaker box. And be careful, because the two big cables coming into the main breakers are STILL LIVE even when the breakers are off.