Can I leave the mosque right after the fard prayer?

The obligation ends with the salam

Once the imam says the final salam and you follow him, your obligatory prayer is complete. Nothing after it is required. The Quran even tells the believers, after Friday prayer, to go out and get on with their day:

So if you're on a lunch break, picking up kids, or rushing to a train, leaving straight away is fine.

What you miss if you rush

Staying a little has real rewards. The Prophet ﷺ said the angels keep praying for a person as long as he stays in the place where he prayed and doesn't break his wudu (Sahih al-Bukhari 445). He himself would stay after the salam at least long enough to say "Allahumma anta as-salam wa minka as-salam, tabarakta ya dhal-jalali wal-ikram" (Sahih Muslim 592). Even a minute of dhikr is worth it. See What should I say after every prayer?

Sunnah prayers: mosque or home?

Many people stay for the sunnah prayers after Dhuhr, Maghrib and Isha. But praying them at home is just as good, or better. The Prophet ﷺ said, "The best prayer of a person is in his house, except the obligatory prayer" (Sahih al-Bukhari 731). So leaving right after the fard and praying the sunnah at home is following the sunnah, not skipping it.

Leaving with good manners

• Wait for the imam's salam. Never leave before he finishes, unless there's an emergency.
• Don't walk in front of people praying. Go behind them or wait for a gap. See Is it haram to walk in front of someone praying?
• Don't step over people sitting in the rows; find another way out.
• Late-comers who are making up missed rak'ahs should finish first; don't rush them.

Arriving early has its own sunnah too. See What is tahiyyat al-masjid?