Programs which run the terminal have two output file handles - standard output and standard error, which have identifier numbers 1 and 2 respectively. "Normal" output customarily goes to standard output, and error messages and warnings typically go to standard error.
When you re-direct output using > or >>, this re-directs only standard output (the former truncating the target file first, the latter appending to the file).
In Borne style shells (including bash, which is a very common interactive shell on Linux systems), you can re-direct standard error using 2> and 2>>. For your purposes, you probably want to re-direct them both to the same file. You can do this by saying "2>&1", which means "take output on standard error and send it wherever standard output goes". This is how:
make > make_output 2>&1