Travelling with pets is difficult, and how best to do it really depends on where you'll be staying and how you'll be getting there. A lot also depends on your pet's personality and preferences. For example, my dogs are fine in the car, but totally stressed out when we stay in a hotel. Pets can also get seriously carsick, too stressed to eat, and can also be exposed to other people's pets' diseases in the rest areas and dog parks along the way. They'll also cramp your style should you want to visit attractions or eat at a sit-down restaurant along your route.
Flying with your pet in cargo is a safe way to get your pet to your destination, though it is not inexpensive. If the temperatures are extreme, you may need to make special arrangements or get special dispensation from your vet. Airlines will also simply not allow you to fly your animal if the temperature in the cargo hold is too hot for your pet to fly safely. I don't recommend carrying either cats or dogs onto a plane, it is likely to annoy other passengers and the size of carry-on kennels is too small for most pets to be comfortable anyway.
Road tripping with you pet really varies in viability, but you absolutely need to make sure that your hotel allows pets before you set out on your drive (La Quinta is very pet friendly, and reasonably priced). You should also research a few vets along your route in case your pet gets sick, in a dog fight, or otherwise injures themselves along the way.
I've done thousands of miles with my 3 dogs in the car, but it isn't easy keeping them happy while cooped up. Riding with your dog in a crate is the safest way to make sure they're secure, and can't escape or interfere with your driving. I don't recommend road tripping with cats unless absolutely necessary, and if you do, the cat must be 100% contained in a crate or on a leash at all times. You'll also have to offer food and water on a thoughtful schedule, so that you're not constantly stopping for pee breaks or getting up in the middle of the night to let your pet out for a midnight poop.