Feature selection for data with both continuous and categorical features?
I recommend using the Lasso for feature selection. I think the main advantage is that it is a multivariate method. However, it is more suited for linear models than for tree-based ones. 

There are some broad theoretical and practical resources that explain why it is used. I wrote one article comparing it to univariate methods and another one comparing it to feature importance.

You can use the glmnet package in R or use the Lasso() class in scikit-learn.