Is it obligatory to perform the two rak'ahs of Tawaf after both the Tawaf of Ifadah and the Farewell Tawaf, and do I bear sin and expiation for having omitted them out of ignorance?

The two rak'ahs of Tawaf are prescribed after every circumambulation. They are a Sunnah (recommended act) and not obligatory according to the majority of scholars. Whoever omits them incurs no sin, but misses out on the virtue. It is permissible to make them up even if a long time has passed between them and the Tawaf.