How can I tell what I am allergic to and how can I get rid of itchy and gooey eyes?

Allergen testing. Can get a complete workup by an allergist, e.g. Skin testing. Otherwise, can try exposing yourself and figuring out the causative agent, removing it, then re-exposing to see if the allergy is confirmed.
Systematic testing. Removal and rechallenge of various potential allergens is the most effective way of determining the causative agent. Sometimes coordinated testing with an allergist can be helpful. At least start by seeing your ophthalmologist - it may not be true allergy...