First off let me sincerely console you for your loss. Our pets are important members of our families...and it's good that you care deeply about your pets.
If the predator were coming onto your property and killing livestock and threatening pets and people...then you have a beef. But putting out traps and making sets in suburbia, amongst your neighbor's and their pets...bad idea.
Coyotes are not hard to trap using snares and foothold traps. I've literally taken 100's of canines, fox and coyotes both. But under your conditions, with no experience at trapping...the odds of you catching catching a coyote are low. The odds are quite high that you'll probably upset your neighbor's.
More importantly, if there are any anti-trappers around you, either you or your traps will just wind up on the evening news and make people like me look bad.
There are lots of things that will kill a cat. Some of them have 4 legs, some of them have 2 legs and some have wings. Unless you actually saw what happened...it's all theory and you have no real idea what happened to your pet.
Contact animal control and file a report.