To remove cable insulation and not damage wire you need dedicated Wire Stripper. There are a lot of different designs of the same tool but i personally prefer something like this one:
To connect wires in most accurate way i can suggest use these little tubes. Put wire ends inside and squeeze with any hand tool. Google something like "50Pcs Non Insulated Rolled Butt Crimp Connectors Terminal For 18-22 AWG Gauge Electrical Wire Cable Connection"
There are also similar connectors for larger size electrical cables
For insulation use tape or Heat Shrink Tube