Avoid those frantic chases before visits to the vet, manage multiple cats (or dogs) and foster cuddle time.

[edit] Steps

Begin by seeing your pet as a living being. Say to yourself: that creature has a skeleton, a beating heart, eyes, and it learns how to interact with you by how you interact with it.
Notice their character traits. This is especially important with multiple animals. Most of the time they have very different ways, likes and dislikes, styles, and they respond differently to requests and affections.
Put aside any silly supernatural notions and realize that you genuinely want to actively hear from them. You won't start to hear them speak, but they are more mentally present and responsive than sometimes assumed.
Stay calm and level with the animals as though they can really hear what you're saying (because, in some way they can!) when it's time to go to the vet or administer some meds. They know something is up, and nervous aloof energy increases the chances of mayhem.
Contrary to common belief, cats can be trained. It is easier the younger that the relationship begins. A combination of positive reinforcement and non-physical messages of disappointment and/or urgency when the act is not carried out will work best. Be consistent with the signals you use to identify and reward a request. That includes key words and phrases as well as visual cues. Most of the time this practice brings the owner and animal into a closer, more mutually respectful relationship and eliminates some of the minuses to having a pet. Keep this in mind when considering giving a pet to an elder or family with small children.

[edit] Tips

Sometimes, it is better to forget the baby talk and reconsider the character of the animal. Sometimes, when used as a tool of persuasion, it works to the contrary and deters the animal from following your request.
Also, many animals feel safer and calmer in cozy spaces like a carrier or even a laundry basket with some blankets (for an inexpensive quick-fix) when traveling or dealing with a crowded house.
No matter what it looks like when you leave your pet, they really have a good time at the vet. I know from experience after it leaves it's owner the dog perks up and is fine. They seem extra excited when you get them back and it seems like they had a horrible time but they probably didn't.