How many different strains of yeast infections are there? What are they called. What type of tests should I have my gyn run?

Ask gynecologist. Your gynecologist will know what types of tests to run, if he/she knows your symptoms and questions. First rule of thumb is to be as honest as possible with your doctor.
Candida albicans. Yeast infections are generally all caused by the same fungus: candida albicans. While there are likely multiple strains of this fungus, they all respond to the same types of medications. Occasionally resistant strains emerge, in which case stronger medications are used. See your doc and follow their instructions and you should be fine.