Is it possible to self install a grounding rod?
Yes, you can install a ground rod yourself.  They are typically 5/8"-1/2" thick 8' long copper rods, which need to be driven entirely into the ground.  It's common to use a large hammer drill, or small jackhammer, with a ground rod driving bit, to drive the rod.

However...  Driving and connecting a ground rod, is not all there is to grounding an electrical system.  If you really want to do this work yourself. Find a person with knowledge and experience with this, that is willing to teach you. If you can't find somebody, go get a couple books on the subject. Study the books cover to cover, and make sure you understand the information before you begin work.

And no, you do not connect things to the plumbing to ground them. Though you can sometimes use the service plumbing as a grounding electrode, and you will have to bond any metal plumbing within the building.

Grounding isn't about connecting to the earth (dirt), it's providing a safe, low resistance path back to the source (service). While ground rods (and other electrodes) are part of the grounding system, they are not the most important part of the system.