It looks like the other answers are assuming that you have multiple lines of data in a file, which are all to be processed. In this case sed or awk (or possibly cut) would be the best tools for processing all the lines in one go.
However, if you just have one line in a shell variable (assuming you're using bash), you can use shell expansions to achieve the desired result, without having to spawn utilities in external processes:
$ var="interesting/bla/blablabal/important"
$
$ # everything after the first slash:
$ echo "${var#*/}"
bla/blablabal/important
$ # everything after the last slash:
$ echo "${var##*/}"
important
$ # everything before the first slash:
$ echo "${var%%/*}"
interesting
$ # everything before the last slash:
$ echo "${var%/*}"
interesting/bla/blablabal
$
Alternatively, I assume your slash-separated strings are file paths. If that is the case, you can use dirname and basename to get the path and filename components:
$ # everything before the last slash:
$ dirname "$var"
interesting/bla/blablabal
$ # everything after the last slash:
$ basename "$var"
important
$
$in the regex.cut -d/ -f2- <infilecutwas designed for this job (extracting one or more fields) whilesedis a "general purpose tool" (so not optimized for this task) ; also, regex is expensive: if you had to process millions of records with hundreds of fields each you'd see the difference...