How do I remove the file suffix and path portion from a path string in Bash?

Pure bash way:

~$ x="/foo/bar/fizzbuzz.bar.quux.zoom"; 
~$ y=${x/\/*\//}; 
~$ echo ${y/.*/}; 
fizzbuzz

This functionality is explained on man bash under "Parameter Expansion". Non bash ways abound: awk, perl, sed and so on.

EDIT: Works with dots in file suffixes and doesn't need to know the suffix (extension), but doesn’t work with dots in the name itself.

The basename does that, removes the path. It will also remove the suffix if given and if it matches the suffix of the file but you would need to know the suffix to give to the command. Otherwise you can use mv and figure out what the new name should be some other way.