Ok, first off, it looks like water is getting into the panel as evidenced by the rust/cancer in the bottom and on the main feeders in the top right breaker. U should first figure out how to keep the water out AND clean up the feeder wires. Condensation maybe? Is the garage fed by overhead service from the house?
Next, a double pole breaker in a 240v/120v fed panel will give u 240v potential between the 2 breaker lugs. So yes, u would run 2 blacks or a red and black to a double pole breaker and a green ground wire to the neutral bar since it appears that theres no ground bar. Which leads me to a question- is this a setached or attached garage? And IF detached, is there any metal running between the garage and the house such as gas or water lines?
For the lift, i would use 12/2 NM(yellow Romex), and put on a dedicated circuit(its own breaker).
Do not share the 220 50 amp circuit for the welder with the lift. Get a two pole 15 for it and mount it in the two spaces on the lower right. Properly protecting a device with the correct sized over current protection is extremely important. Long explanation short, installing a high ampacity breaker on too small a conductor is how fires can start. As for the other 120 circuit you want, pick up a twin breaker. These have two switches in a single pole size breaker.
Re-reading I may not have answered your question. That 50amp, the top pole is connected to phase A and the bottom to phase B, going across phase, A to B is where you'll get 220. Either phase to ground is 120v. Sharing the compressor and the welder is fine, run #8 wire to both. Your compressor should have a built in over current protection device, most starters do. But fir the lift you only need #14 gauge. The lift will not have a over current device built in. You could however get a two pole 30 amp rated fusable disconnect and install two 15amp fuses and use the shared 50 amp circuit.
Sorry if this came across a bit disjointed, phone posting is tuff for me.
I wouldnt suggest wire sizes to feed a compressor or welder when u dont even know the load or size of the unit. And not all compressors come with starters. Many come with just a pressure switch which has no overload protection.
ok, I think the info i need is in your post, just trying to translate it to something I can understand.
the first part about running a separate breaker for the welder I udnerstand, will do.
So that 50A breaker techincally has 220V in it, all i need to is run my cable from the lift to that?
which colors do I "plug" into that breaker? 2 wires go into the breaker right?
and then the bare copper goes to the bus bar just like the others?
Yes, a double pole breaker has 240v potential between its lugs. And 50a would be WAY to large to feed that lift. BTW, nominal voltage is 240v NOT 220v!