Health and Safety

Health certificates for pets are mandatory when you're traveling by plane, so it's always a wise decision to have your pet receive a health check prior to takeoff. If you're traveling by car, a veterinarian visit may not be as necessary. You should always make sure your pet is healthy and happy before loading up the car. It can be very stressful and distracting when you're taking a long road trip with a sick pup!
Identification

Always make sure your dog or cat has proper identification before leaving for your destination! A sturdy collar with an ID tag containing the pet's name and the owner's phone number is a must. Also, it's a good idea to bring a spare leash just in case something happens to your main one. You never know what the future brings, which is why it's always good to be prepared. Another great form of permanent identification would be a microchip.
Traveling by Plane

Traveling by plane can be stressful for not only a pet, but for the owner, too. In an effort to make air travel less anxiety-ridden for your dog or cat, an option would be to have your furry family member certified as an Emotional Support Animal. By doing this, your pet is able to fly on the plane sitting by your side, relieving stress and worry. (Note: It's a much easier process to have your animal registered as an ESA if he/she is well-behaved and quiet!) All animals are able to be accommodated underneath the aircraft, but many individuals prefer to keep a watchful eye on their pets.
Stay Calm

Your pet's emotions can easily be affected by yours, so no matter the situation, you should strive to stay calm. No loud voices, fighting or screaming. Pets can pick up on all of that. Difficult situations may arise while traveling, which is why it's important to remember to stay mindful—your pet is traveling with you, too, and you don't want to upset him or her!
Traveling by Car

A lot of pets don't mind car rides, but unfortunately some do. Depending on the length of the car trip, there are different types of medications you can give your pet to make the drive less anxiety-ridden for him or her. There are medications to make your pet sleepy and relaxed, as well as meds to help with nausea.
Many dogs get car sick and throw up, which is why it's always good to have a towel on the seat underneath the dog to prevent a messy and expensive cleanup! Cats should always be contained in a carrier or crate, no matter what distance you are going. They may not like it, though. And when cats are not pleased, the meowing never ends. With that being said, soothing voices telling your kitty it will be OK, and low music, help to calm them.
Also, never leave your pet unattended in a vehicle, no matter how short of a time you might be gone.
Familiar Blanket or Toy

Almost all pets have a special toy or blanket that they sleep with or carry around. Like small children, it is comforting for your pet to have access to his or her toy/blanket at all times when traveling. A small toy that you can stick in your pet's carrier with him or her could totally make the trip more bearable. This would ease your pet's stress—and make it a more enjoyable process for you!
Proper Hydration

No matter what source of transportation you choose, water needs to always be available for your pet. If your pet is traveling by plane, make sure that a water bowl is kept near him throughout the flight. If you are traveling by car, make frequent stops offering your pet some hydration. There are also portable water dishes you can purchase for your pet, like this one from Petco.
Medication

There's a lot of pets out there that need all types of medications for different things, such as diabetes, cataracts, nausea, even flea medication! It can be an easy thing to forget to pack, so whether you need to put a reminder on your phone or a sticky note on the door, don't forget your pet's medication! It can be frustrating and expensive trying to find your pet's meds while on the go. You should also always pack an extra pill or two. You never know if you'll have to stay another night.
Travel Kit

Bring food, treats, a bowl, leashes, first aid, medication (if needed), plastic bags and favored toys when traveling. You don't need to go all out and overpack, but a smallish kit that is able to fit all of the necessities for your pet would work well.
Hotel Research

If you're staying at a hotel while traveling, always do your research! Although many hotels and resorts allow pets, many don't, or have an extra fee you need to pay. This is a simple task and just requires a little bit of calling and looking around online. Never assume. You should always be sure that your pet is allowed to stay wherever you are.