My baby's neck measurement was 5.3 mm and they say he might be have down syndrome? What to test next?

Screening vs direct. I'll assume your question is about a prenatal ultrasound finding. These tests are indirect ways to select out cases where a more direct or definitive test would be worthwhile. A prominent neck is suggestive but not definite.The more invasive test, an amniocentesis takes a sample of the fluid in the womb that would have baby cells. That could be tested for the ds chromosomes & know for sure.
Chromosomal studies. If your child displays some or all of the characteristics of down syndrome, your doctor probably will order a test called a chromosomal karyotype. This test is an analysis of your child's chromosomes. If there's an extra chromosome 21 present in all or some of the cells, the diagnosis is down syndrome.