I'm not a doctor, and this is not to be considered medical advice.  People always ask me how do they find out if they are gluten or dairy intolerant.  Here are the typical tests I am aware of, and I've been told that it is a good idea to ask for all of the tests.

Serological Tests for Celiac Disease
IgA AGA
IgG AGA
IgA tTG
IgA EMA
Gliadin AB IgG
Deamidated gliadin peptide

Further Testing
Scoping / Small intestine biopsy
Genetic Testing - to see if there is a pre-disposition to the disease.
Allergy testing
Biopsy for DH

Serological Tests for Dairy Intolerance
IgE (Milk-specific immunoglobulin E for dairy allergy)
IgG
IgA

Further Testing
Hydrogen breath test (for lactose)
Lactose tolerance blood test
Allergy testing