from your description its looking like to be a bacterial conjunctivitis
as it has spread to the other eye so fast and stickiness of eye is there. u can start with antibiotic drops like TOBRAMYCIN EYE DROP
, 3 times to both the eyes, can continue for 4-5 days and clean the eye with luke warm soaked cotton balls. try to prevent rubbing of the eyes.once the infection is controlled, the swelling will go by itself.