How easily are std's transmitted to the child during birth?

Depends on the STD. Some pass during pregnancy (syphilis), some rarely pass at all (HPV). Gonorrhea can pass during the birthing and trigger blindness. Chlamydia can pass during birthing and cause pinkeye but also not show up for weeks when it then produces a terrible pneumonia. Recognition and treatment during pregnancy can stop any of this from happening.
Gonorrhea certainly. Gonorrrhea can cause blindness in babies that deliver through an infected vagina. Syphillis can be transmitted through the mother's blood stream in utero. Every pregnant patient undergoes some form of std screening by the OB md for that reason.