One possibility (based on an original suggestion by Kyle Meyer) is
(global-set-key (kbd "C-c m") 'recompile)
Then you can type
C-c m, which will first offer to save the current buffer if necessary, and then will run the command
make -k in the directory containing the associated file. The
-k switch causes make to try to keep going even if some commands fail. In case you want to modify this, you can type
C-u C-c m and you will get the chance to edit the compilation command in the minibuffer first. The edited command will then be remembered on a per-buffer basis.
Any compilation errors will be shown in a dedicated buffer (called
*compilation*) and you can either click on these with the mouse (to jump to the relevant point in the source file) or use
C-x backtick to cycle through them (sorry, I can't work out how to type a literal backtick inside backticks!). The compilation buffer is not shown if no errors occur but you get a notification in the echo area when compilation is complete.
Of course, you can use any other key of your choice instead of
C-c m - I chose that for the mnemonic
m = "make" and because the
C-c letter keys are reserved for the end-user, so there is little risk of collision.