As a vegetarian, what blood tests should I have done to make sure that I am getting all of the nutrients that my body needs?

Here's my suggestion. You know you need B12 checked. Montior iron and vitamin D levels. Truth be told, your best bet is to get with a vegetarian-friendly evidence-based physician to guide you. Best wishes.
I can think of three. As a true vegan ( no dairy or eggs), one may be deficient in iron, vitamin b12, and vitamin d ( although non-vegans often also lack vitamin d). Let your doc know the extent of your diet and how long you have been a vegan. This will help him/ her in selecting appropriate tests.