Trapping is a great hobby to take up and I believe its an important heritage for us to keep alive as sportsmen. The best way to learn about safe, responsible and effective trapping is to enroll yourself in a trappers education course with a local (or national) trapping association. We're fortunate to have numerous educational opportunities for young & aspiring trappers in the Midwest, but I'm not sure how hard they are to find on the East Coast.
Non-Target catches (IE: Domestic Dogs) are not entirely avoidable but some simple techniques and practices can GREATLY reduce the likelyhood of snagging something you didn't intend too. Whether you're using foot-hold traps or snares, domestic canine's are generally very easy to release unharmed. I wouldn't let the fear of grabbing a non-target animal keep me from taking up the sport of trapping!
In my region we don't have many large sprawling fields but when you find one you'll notice coyotes are some of the only critters that working those fields over. Sets made a good distance from treelines and woodlots will often prevent non-target catches like possums, raccoon and domestic animals of all kinds. With some experience in trapping, calling and scouting coyotes... you'll learn that there are three fundamental keys to catching coyotes. Location, Location, Location. Of course there are a variety of stratagies for selecting equipment, bedding traps, placing lure and blending sets... but location is king.
Here are some pictures just for kicks-