IF a cell contains a string

You can use OR() to group expressions (as well as AND()):

=IF(OR(condition1, condition2), true, false)

=IF(AND(condition1, condition2), true, false)

So if you wanted to test for "cat" and "22":

=IF(AND(SEARCH("cat",a1),SEARCH("22",a1)),"cat and 22","none")

SEARCH does not return 0 if there is no match, it returns #VALUE!. So you have to wrap calls to SEARCH with IFERROR.

For example...

=IF(IFERROR(SEARCH("cat", A1), 0), "cat", "none")

or

=IF(IFERROR(SEARCH("cat",A1),0),"cat",IF(IFERROR(SEARCH("22",A1),0),"22","none"))

Here, IFERROR returns the value from SEARCH when it works; the given value of 0 otherwise.