What your equation predicts and what actual data look like make up the serial correlation case. So if the plot shows actual data points clusterd above or below the predicted line from the regression, with or without patter, it indicates serial correlation.
Heroskedasticity is the variance of the error term, that is how the errors (predicted minus actual) vary around their own mean. They should show consistent yet random variation. Residual vs. independent variable, as you say, helps with serial correlation, but not directly helpful for heteroskedasticity…however, check with some quant heads, they may know better.