If you need to replace cone gears you're going to have to pull the entire gear box apart. Take out the tumbler shaft first. There is a collar on the inside that has a roll pin you need to punch out first. Then take out the dog clutch shaft. Make good notes and take photos of the orientation and order of the dog gears. The dog clutch slide, rides on a key that is really hard to get out. You need to get all the splines lined up so that the key is between them so you can lift the one side out. There is one snap ring on the dog gear clutch. You may need a press to press some of these shafts out. Finally take out the cone gear shaft. There is a snap ring on that shaft as well. Looking towards the lathe the gear on the left side is integral to that shaft, so there is only one way to press that shaft out. This is by no means comprehensive step my step, just the few key points I'd point out since I've recently had to take apart my entire gear box.
If you need to replace gears I'd suggest buying standard gears and boring/broaching them as needed. If you order from Clausing they will cost and arm and a leg. They are pretty typical spur gears with a 14.5 degree pressure angle.
I had to replace the integral gear on the cone gear shaft and so I ordered a 16 tooth gear from McMaster Carr, machined it to fit, machined off the worn one and pressed on the new one. Saved my about $100 over buying one from Clausing. And that's exactly the sort of project that I got this lathe for!
Let us know how you make out.