What size wire is appropriate for 150 feet from house to shop.Shop 60 amp box. Will run shop lights and a large 220 volt air compressor off box and few outlets. Not all will run at same time. Maybe lights and compressor together. I live in North Carolina near Durham.

I would recommend #4 awg to keep the voltage drop below 3%...

Is the wire size your only question?

Do you know what wire to use?
Are you running conduit?
What about grounding and bonding?