At what age do babies begin to synthesize their own antibodies?
In the graph above the darker blue line refers to the antibodies the baby receives from the mother in utero, as you mentioned in your question.  

As you can see, the red line indicates that babies begin to produce low levels of their own antibodies between 3 and 6 months before birth.  However, these are IgM antibodies, immature 'rough draft' versions.  These have much lower affinity for antigens then their mature IgG counterparts which are the classically thought of antibody.  

Levels of an infant's own IgG start to rise after birth, however don't reach a reasonable level until after the child is roughly 1 year old.  Maternal antibodies start to tail off at around 3 months leaving a period highlighted in blue on the graph where infants are particularly prone to getting infections.