I have searched the internet but could not find specific information so if we have got the following document "Carta di Soggiorno di Familiare di Un Cittadino Dell'Unione" (Residence Card of a family member of a Union Citizen, Italy) issued by the Italian authority, what rights do we possess of working inside Italy and EU? What about travelling? Will we be requiring visas for the states which are are not in Schengen?
Work inside Italy is permitted. The card gives no right to work outside Italy. However, the holder enjoys freedom of movement when traveling with the EU family member, which implies a right to work. Other countries can ask for independent proof of this right before allowing work, but individual countries may be more permissive. Independent of the card, the holder may also be permitted to work in other countries under the freedom of movement of services.
The holder can travel independently within the Schengen area for 90 days out of 180. I'm not sure about the Schengen-candidate countries like Romania and Bulgaria, but the UK and Ireland will require a visa for the holder of this card unless the holder is exempt from that requirement by virtue of holding a passport from a country whose citizens are visa exempt.
Travel with (or to join) the EU-citizen family member
The 90/180 rule does not apply within the Schengen area, though countries may require registration for stays longer than three months. For countries outside the Schengen area, it is not permitted to require a visa. The UK requires proof of relationship, however, and it may also be prudent to carry such proof when traveling elsewhere.