Signing of the Cross

I recently went to my first Byzantine Divine Liturgy as a Latin Rite Catholic. People signed the cross many many times during the liturgy, and I am confused on when and why you should sign it.

Additional question: At the end the liturgy after communion, we kissed a cross the priest held and ate what seemed to be bread before leaving the church. What was this and what should I do next time it happens? Idk if this relates to the ritual, but we were remembering and praying for a dead member of the church that day.