How can I check the first character in a string in Bash or Unix shell?

There are many ways to do this. You could use wildcards in double brackets:

str="/some/directory/file"
if [[ $str == /* ]]; then echo 1; else echo 0; fi

You can use substring expansion:

if [[ ${str:0:1} == "/" ]] ; then echo 1; else echo 0; fi

Or a regex:

if [[ $str =~ ^/ ]]; then echo 1; else echo 0; fi