The Y and X data do not need to be Normal to do regression (neither the Ys nor the Xs, single X or multiple Xs). Normality comes into play when examining the residuals. When the residuals follow a Normal distribution (a Normal Probability Plot is my favorite tool), then the regression model should be sufficient for prediction. If not, then another model or data transformation should be considered.
The P values and R2 values will help to determine the strength of the relationship – there is no connection to Normality.
In addition to testing the residuals for Normality, it is a good idea to plot them against the predicted values and the observed order to see if there is a pattern that is not random.
It is always a good idea to include a prediction interval, for either a point estimate or a regression plot of the data.