What are some good, simple ways to explore triggers for asthma attacks?

Diary. I would not recommend that you simply "explore" on your own, specially if the symptoms are severe. Keeping a detailed diary on patterns of flareups, should help you identify what may be your triggers. Keep in mind that some of the most common are: allergens like pollen, dust mites, mold, pets, and exercising. Allergy testing would also help.
Allergy testing! The primary triggers for asthma include infections (common cold), allergies, and exercise. Allergic triggers can be grouped into two categories: seasonal (pollen) and perennial (pets, dust mites, and cockroach). Allergy testing is a simple, accurate, and cost-effective way to diagnose such triggers. I would recommend evaluation by a board certified allergist for this testing.