I would like to run an prediction model and have a set of continuous independent variables. They are all important but highly correlated. How can I effectively reduce collinearity and still use these variables in my prediction model?
You can reduce multicollinearity using PCA. There are lots of questions/answers about how to implement PCA. This method allows you to group similar covariates into independent "Principal Components" which can give insight into the relative relatedness of your covariates.
Also, check into Variance Inflation Factor (VIF) protocols. There are ways to use stepwise VIF reduction to rid yourself of highly collinear variables in the dataset. However, if you need to keep every covariate for some reason, a clustering approach like PCA or PLS would do the trick.