There are a few places that require metal drain pipes, so before you try to convert, you need to verify what is allowed. Then, while you'll have to sign your life away, recycle the copper, it's worth a lot of money per pound, it will help pay for a bunch of your new materials! Since there are lots of thieves stealing copper out of buildings, you have to prove to them it's actually yours to recycle.
Keep in mind that what was to code in 1960, may no longer be, and you may not be able to just try to replicate what you have in say PVC. When you make significant changes like replacing all of the pipes, whatever is done must be to current code. You need to check which code your locale uses, as not all places use the same one, some are more lenient than others as to what is allowed and how to do it.
It looks like at least that CI drain line is still in decent shape, so the conversion from copper to plastic would start with removing the copper in the hub. Then, you'd measure the cleaned up opening after removing all of the lead and oakum, pick up the proper rubber donut seal, and insert it and the pvc in and start building up (easier than working from the top down). Since you won't get it all done in a day, you'd want to pick up some banded couplers (you'd need the right one to convert between pvc and copper) so you could reconnect things as you go.
This may not be a DIY'er friendly job, as the plastic pipe has thicker walls, and you may need to make the holes larger as you go through things, plus, it will take awhile, and I assume you plan to live there while it is happening. Plus, as mentioned, the code changes that have occurred in the last 50+ years.
Without seeing the entire setup, there is no way to tell whether you can just duplicate what's there, and what's there may not be the most expedient way to replace things. There are likely more fittings for pvc than there were for copper, and you'd have more choices.
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