I have a carta di soggiorno di familiare di un cittadino dell'unione. I am Pakistani. Can I visit Schengen countries without showing my Pakistani passport or without going to the Pakistan embassy?
Can I visit Schengen countries without showing my Pakistani passport[?]
Probably not. It depends on which countries you're going to. In general, Schengen countries can require you to have a passport in addition to your residence card, and many of them do indeed have such a requirement. You should take your passport with you.
... or without going to the Pakistan embassy?
Yes; there's no need for you to go to the Pakistan embassy unless you need to get a new passport.