Do high VIF values for a a particular variable $x$ just indicate that it is highly correlated with at least one of the other variables in the model? Does it specify which variables and how many variables that $x$ is correlated with?
No. That's one reason it is better to look at condition indices and the variance proportion matrix. In SAS PROC REG you can get this with the /collin option on the model statement.
See Regression Diagnostics by Belsley, Kuh & Welsch or Conditioning Diagnostics: Collinearity and Weak Data in Regression by Belsley (out of print, but more thorough on the problem of collinearity specifically). Or you could even look at my dissertation: Multicollinearity Diagnostics for Multiple Regression: A Monte Carlo Study (haven't really looked much at this since then, though).