I have documents that runs tens of typed pages dealing with this topic...its anything but simple.

Some starting points http://www.listendata.com/2015/03/ch...-multiple.html

http://people.duke.edu/~rnau/testing.htm

For Heteroscedasticity the simplest is just looking at the residuals to see if there is a pattern, the spread, getting bigger or smaller...

For non-linearity look at partial regression plots and see if there is any obvious non-linearity.

For normality run a QQ plot for the residuals against a normal distribution.

Running tolerance checks for multicolinearity.

I am not aware of any formal test for non-independence except for that which occurs in autocorrelation. You can test for that with test such as Durbin Watson or you can look at the ACF in ARIMA (the former is much easier).

I have a tome of how to do this in SAS, but no way to send it unfortunately. Not that model fit/value which is why you seem to be talking about is different than the test of the assumptions I make above.