I'm shipping hundreds of important packages a year - I don't want headaches.
For cues, I always use FedEx 2 day in their triangle box if it fits, insured. I have a commercial discount with them. My cost is usually $20 or so and i do it all online. I had one package damaged by Fed Ex, filed a $400 claim and was paid within two weeks. I wrap the cues in paper or thin bubble wrap so they don't move around in the box, and put JP's on them if I can.
My experience is UPS is rougher on packages and also had glitches in their tracking system, so they are second place.
USPS is a distant 3rd place. I've had cues snapped in half, run over boxes, etc, etc., delayed, untrackable if addressed improperly. Their service is spotty. If something does go wrong, it's much harder to straighten out than Fed Ex or UPS. Their express mail is excellent, especially internationally (which I do use and and actually prefer for my own reasons) but then you're into Fed Ex prices.
Chances are all of these providers will provide adequate service, but in my experience, when it comes to quality cues or antiques, don't cheap out - use the best, it's just not worth it.
Chris