Our home has two prong outlets with no ground wire and no ground opening in the outlet. I think I can run a separate copper xx 2 wire to each outlet and have them go to either a water line (grounded) or to the panel ground bar. I assume I could pigtail these together so I don't have 20 lines? How should this be done in the most cost effective way (assuming I can do the work).
Thanks!

This would not be to code and illegal, unsafe. Another option is to install GFCI outlet which will afford you 3 prongs but will not be truly grounded but will be ground faulted. Grounds wires must be in the same conduit or cable as your black and white wires.

Thanks. Okay, so I can put in GFCI's that are protected but not grounded (or ground faulted). Looks like that is the least expensive fix. I'm also reading that this is not suggested for electronics. Should outlets that electronics are plugged into have the wiring changed back to the breaker box so that it is ground fault protected and has the third load wire in the cable?
For clarity.. . Someone told me a while back that you can ground a single outlet for a computer by running a ground wire from the outlet to the grounding rod (hopefully close by). Is that ill advised and dangerous as well?

"Someone told me a while back that you can ground a single outlet for a computer by running a ground wire from the outlet to the grounding rod (hopefully close by). Is that ill advised and dangerous as well?" omebody told you wrong, this is not allowed. What you can do, if you have space/room to run that wire, is to runa cable with your black,white and ground directly to the panel from the outlet.