I assume the ground going to a mix of PEX and copper is not good. Therefore, what should I ground to down in my basement (1973 house)
A 73 home may have a separate ground rod. It is usually out side of the home, near the electrical service. The top of the rod may or may not be exposed. Look for a #6 wire (may be bare copper or insulated). Modern NEC requires 2 ground rods, 8 ft long, driven all the way into the ground, and spaced at least 6 ft apart. Please note that local codes often modify this and your area may have different requirements. If you have a ground rod, this is considered as the main ground for the home. The 'ground' wire that goes to the water pipe is actually referred to as a bond wire. If you are only replacing some parts of the copper pipe, then you will need to buy pipe clamps (like the one that you see connecting the wire to the pipe) and some bare copper wire (#6 if you have a 100 amp service, #4 if it is a 200 amp service). You will need to make sure that you install 'jumpers' everywhere that you eliminate the copper pipe to keep the remaining sections of copper pipe electrically connected. If you are eliminating all of the copper pipe, you will no longer need the bond to the pipes (Make sure that you do have a ground rod). This will also apply if the only remaining sections of copper are the short pieces that are used only for connecting the plumbing fixtures. If you have to drive new ground rods, it must be the current NEC (or local codes). Clarification: If you do not have a ground rod and you are eliminating the bond wire, you will need to add ground rods. You can probably drive the rods yourself, but I would recommend a pro for the wire connections.
both sides of the copper pipes that the plastic connects
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Ground to either side of the copper pipe, bypassing the plastic pipe to the copper pipe and use 6mm cable. Keep ground continuity at all time or any fault current will go through you and not the ground.