How to search for one of two similar strings in Vim?

/planet\(Awesome\|Terrible\)

To see the relevant documentation, issue :help /, and scroll down to the section “The definition of a pattern”.

To add to Gilles' answer, you might want to add a few things in there:

/\<planet\(Awesome\|Terrible\)\>

\< marks the beginning of a 'word' (essentially alphanumerics)
\> marks the end of a 'word' (essentially alphanumerics)