Use the set -e builtin:
#!/bin/bash
set -e
# Any subsequent(*) commands which fail will cause the shell script to exit immediately
Alternatively, you can pass -e on the command line:
bash -e my_script.sh
You can also disable this behavior with set +e.
You may also want to employ all or some of the the -e -u -x and -o pipefail options like so:
set -euxo pipefail
-e exits on error, -u errors on undefined variables, and -o (for option) pipefail exits on command pipe failures. Some gotchas and workarounds are documented well here.
(*) Note:
The shell does not exit if the command that fails is part of the
command list immediately following a while or until keyword,
part of the test following the if or elif reserved words, part
of any command executed in a && or || list except the command
following the final && or ||, any command in a pipeline but
the last, or if the command's return value is being inverted with
!
(from man bash)