I was 37 and having my first baby--a fairly high risk situation. One of the tests showed possible Downs, but I don't remember which one it was. My husband and I opted for no more testing, especially the amnio, because we would not have terminated our pregnancy regardless of the outcome and we had taken 7 years to get pregnant and did not want to hurt our baby. We considered our child a gift from God, and if He was going to send us an "angel" baby, we would do our best to take care of him/her, regardless of his/her condition.
Due to my age, we did have a couple of ultrasounds. Our beautiful daughter is 15 years old now. She is an straight-A student taking weighted classes.
Ask lots of questions and do your research. Knowledge is power. I think the geneticist sounds like a really good idea. Blessings to you and your child.