Your dr might start with blood tests to see if you have high levels of certain antibodies showing you may have a sensitivity to gluten. If positive, they will likely want to do a biopsy, as below. Blood tests are not as reliable as the biopsy for dx.
The "gold standard" for a celiac dx is the intestinal biopsy showing damage to the villi in your small intestine. That involves getting a gastroscopy done, where a tube is inserted from your mouth to small intestine. You can have this test done with or without the sedation, but I would recommend getting the sedation, you will remember little to nothing about the procedure and it's fast, a gastroscopy is about 15 - 20 minutes in length.
You can get genetic testing done, which can show you have the genetic tendency to develop celiac.
Your symptoms do sound like you might have celiac.
Some people just go GF and see if they feel better, as I did after basically guessing myself. I had allergy tests done that showed I was allergic to wheat, rye and barley and that's when the light went off for me and I went GF at that point in time. Problematically, it makes getting a dx much harder as the levels of antibody reduce in your bloodstream and damage to your villi begins to heal. SO, if you are going to try to get a dx first, DO NOT stop eating your regular diet.

to you - I used to have to know where every bathroom where I was shopping or in the grocery stores, so I know it's challenging and not fun to go through all this. Plus, it's likely my thyroid problems and other issues were related for years, including poor enamel on my teeth.
A good link with concise and up to date info:
http://www.celiacdisease.net/overview