First, I would check the state and local laws and regs. A .223 will do just fine. A 6mm Rem will give you some extra range.
Again, check the laws and regs in your area. If electronic calls are legal in your area, you can download free MP3 files from varmintal.com and play them on a portable boom box. You don't need to spend a lot of money on an electronic caller (if allowed). The hurt pup or rabbit in distress can be very effective. Some areas will also allow baiting, but some areas so not allow baiting.
If you have some very cold weather or a storm, they will be out more after these times looking for food. If your area allows hunting at night, try hunting just after dark. If night hunting is allowed, be sure to follow all rules and regulations regarding the lighting allowed, etc. Also be sure of your target, what is between and what is beyond.
There is no shortage of coyote hunting videos on Youtube.