I have a black wire a white wire and a ground wire coming into plug. Every wiring diagram keeps saying I should have a red wire. I don't. This is a old house and I'm needing help with hooking up wiring plug mounted on wall and help on how to wire into panel box.

You don't have to have a red. The white wire should have been tagged red with tape. Wouldn't that have ended some of the confusion.

A 4-wire dryer cord is the new standard and it makes a lot of sense. It would be nice to re-run the correct wire if it isn't too difficult.

The Dryer's instructions should mention a ground bar. You need to bridge ground an neutral at the dryer and connect to ground.
Black is one hot and white is the other for a 2 wire w/ground circuit. There is plenty of resources on the net and even here.