Fantastic thread! Sadly something often overlooked, especially with first purchases. I always suggest clients establish a relationship with the vet before bringing the new horse home.
I love my vet, have an awesome working relationship. I've been with her since she started her practice. When you develop that kind of relationship with your vet, you can sleep well knowing that your horses will be taken care of when they need it. I'm lucky that I've got a good enough relationship with mine to have her cell & unlisted home phone numbers, no going through the answering service in an emergency. When we have mares in foal, she keeps track of due dates and will call just to check in on them as we are getting close. She called a couple days ago to check on my old man, we've had above normal temps and no rain, it's been miserable. She just wanted to make sure he wasn't wilting in the weather. I also return the favor for her, there have been a few times when her assistant has been sick that I've rode along for farm calls to help her out and I've also taken care of her horses when she is on vacation. She notes it on my account and gives credit whenever I have a bill.
When mine goes on vacation, I have 4 others that I can call in an emergency and a university vet school within an hour drive.
Excellent point made earlier about having the numbers handy, mine are all in my cell phone and I've even made vet & farrier some of my emergency contact numbers. If I have an accident on a horse that should require me to need assistance, it's likely my horse will too.
Life is like a camera. Focus on what's important, Capture the good times, Develop from the negatives and if things don't work out, Take another shot.