do I need a Schengen visa...
No, you do not. Your Carta di soggiorno is a "residence permit" for the purpose of the Schengen Borders Code. You do not need a visa in addition to that card.
This is at Article 2(16)(a) and Article 6(1)(b). Article 2(16)(a) (emphasis added):
‘residence permit’ means:
(a) all residence permits issued by the Member States according to the uniform format laid down by Council Regulation (EC) No 1030/2002 and residence cards issued in accordance with Directive 2004/38/EC;
Article 6(1)(b) (emphasis added):
For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, which entails considering the 180-day period preceding each day of stay, the entry conditions for third-country nationals shall be the following:
(b) they are in possession of a valid visa, if required pursuant to Council Regulation (EC) No 539/2001, except where they hold a valid residence permit or a valid long-stay visa;
Furthermore, the card makes you a "person enjoying the right of free movement under Union law" under the code, so you can use the EU/EEA/CH lanes even though you do not have an EU, EEA, or Swiss passport.
This follows from Article 2(5)(a) and Article 10(2). Article 2(5)(a) (emphasis added):
‘persons enjoying the right of free movement under Union law’ means:
(a) Union citizens within the meaning of Article 20(1) TFEU, and third-country nationals who are members of the family of a Union citizen exercising his or her right to free movement to whom Directive 2004/38/EC of the European Parliament and of the Council applies;
- Persons enjoying the right of free movement under Union law are entitled to use the lanes indicated by the sign shown in Part A (‘EU, EEA, CH’) of Annex III. They may also use the lanes indicated by the sign shown in Part B1 (‘visa not required’) and Part B2 (‘all passports’) of Annex III.
Speaking of passports, you will need your Indian passport in addition to the carta di soggiorno. (That's in Article 6(a) and in the freedom of movement directive, but I'll refrain from quoting them here.) Therefore,
...or can I board my flight with my Italian family card?
Not by itself; you'll also need your passport.
Some may say that you cannot benefit from freedom of movement in Italy because, as the spouse of an Italian, your relationship with Italy is governed by Italian law rather than by EU law. But Italy has passed a law that extends any benefit of the freedom-of-movement directive to the family of Italian citizens.