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

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.

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.