Car now cranks over and runs, but electrical system goes haywire?

Yesterday, I asked a question about my car not starting. It now starts, as the ground wire was bad and one of the relay cords had burned off of the starter. Those were fixed and my car now starts and runs. I took it for a test drive last night (and this morning) and the dashboard does a funky thing. After about 20 seconds of driving, the dashboard lights will all turn on. Sometimes the RPM meter and thermometer will fall completely down even though I am running the car and driving. Then everything will go to normal for a few seconds, and then repeat the process. There is also an audible click when this happens inside of the hood around the area where the battery/starter is at.

I just got a new battery and new starter if that matters. What could be the cause of this, and is it something as simple as tightening a few bolts and cleaning other things?