What is the appropriate data to display with linear regression? What should be included on a graph with a linear model with a significant p-value? 
I see a lot of different regression graphs displaying everything from Pearson correlations to the linear model itself to just $R^2$. Is there a basic "minimum" that must be included on a linear model plot?  What would the ideal graph display?
 A: As a bare minimum you need to plot the points and the line. If there is substantial overprinting so that the plot becomes misleading consider adding some jitter (random noise in the $x$ or $y$ direction) or using sunflowers or similar.
If someone also working in your field comes along and wishes to compare their results with yours then they will need to know the values of the coefficients with their standard errors. You could put this in the caption or even have a separate table for the additional information. Whether it is worth including measures of fit like $R^2$ varies from case to case. It may well be that finding a relationship is very important even if it accounts for little of the variance.

Note that this answer is about a linear regression and the desire expressed for a single graph. If the regression is in fact a multiple linear regression or the desire is to have more than one graph then @Ben's answer is the relevant one.
A: I am going to consider the general case where you may have multiple explanatory variables in your linear regression, in which case it is often not feasible to plot all the data with the fitted regression (hyperplane) through the data.  Moreover, while it is true that you can look up published papers for examples, my experience is that many published papers do not show good graphical outputs, and so this might simply lead you to copy bad habits.  In any case, once you have obtained your final model, you probably want to show various things with your plots, and I would recommend using some or all of the following:


*

*Show that model assumptions are reasonable: This would be done with diagnostic plots, including the residual plot, residual-scale plot, residual KDE (superimposed against T distribution), residual QQ plot (against the T distribution), leverage plot, and residual-leverage plot.  These plots would usually be relegated to an appendix, with brief discussion in the body of your paper.  This allows the reader to see that the assumptions underlying your model are reasonable (or not), and make inferences accordingly.

*Show relationships between an explanatory variables and the response: The way to do this is with added-variable plots.  An added-variable plot is a scatterplot of the two variables that first filters out the effects of the other variables in the regression, so it gives a valid representation of the conditional relationship in the model.  One salient benefit of this plot is that the line-of-best-fit in the plot has a slope equal to the estimated regression coefficient for that variable, so it is consistent with your regression output.  In my opinion, you should not use an ordinary scatterplot of the two variables --- at best it is worthless, and at worst it is actively misleading.

*Visualise the information in the regression model outputs (optional): This can be useful in cases where you want the reader to be able to visualise some aspect of the regression output.  The appropriate graphs depend mostly on the number of explanatory variables you are using.  If you do not have too many explanatory variables to make it infeasible, you can summarise the coefficient estimates graphically with a simple barplot, with error-bars showing confidence intervals.  (I recommend you put your variables on an appropriate measurement scale to allow easy comparison of the relative size of estimates.)  Similarly, you can summarise your ANOVA graphically either with a barplot, or even a pie chart (though some statisticians counsel against using the latter).  These types of plots would typically go in the main body of your paper to assist the reader to visualise some aspect of the regression output.  (The actual regression output is usually in an appendix or supplementary material.)
