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Lots of people have pets in USA. Most popular are dogs and cats. Therefore many public places accept pets, like restaurants (but only outside), some hotels (but they might charge pet's fee) and parks (but pet's have to be leashed all the time). Cities have dog's parks where dogs can run and play together. There is also plenty of stores where you can buy all sorts of things your pet might or might not need.
Travelling with a dog is difficult. Only small ones and service dogs are allowed in public transport. Museums and galleries do not accept dogs. Also many national and state parks do not accept them, especially on the West Coast (and those are the most beautiful parks).
People are generally friendly towards pets and people with pets, but Americans are very protective when it comes to their land, so make sure you keep your dog on leash, clean up after him/her and keep your pet off somebody's property.
Vets are ridiculously expensive In USA. I heard stories about dogs getting cancer and owners spending 40 thousand dollars to prolong their lives. Even if your pet is healthy and in good shape visit at the veterinarian will cost you 100 dollars. Once my dog ate some resins which can cause liver failure. I rushed him to the vet where they took very good care of him, but I paid nearly 500 dollars for half an hour visit. Doctor suggested that I might want to put my pet to the hospital for 48 hours so they can monitor his health. That would cost me 2000 dollars.

I agree to Polrad's comments, made the same observations. Bringing cats and dogs with you is relatively easy if they have the required vaccinations.

I have always had dogs when I lived in Cape Town, South Africa, but the culture and climate in New England, USA require a lot more care and vigilance over your pets.  I have a pug called "Whiskey" who has to be treated, groomed, fed with special food, and has regular vet visits. It is advisable to have pet medical insurance, or the visits to the vet could add up to a few thousand dollars per year. Pets are indeed integrated into society, and attending a "Doggy Day Care" is important for socializing skills. Your pet dog becomes a real family member as opposed to my SA experience where my dog would live outside in a kennel. My church has an annual "Blessing of the animals" worship service where you can experience about 30 well behaved dogs who would quietly sit throughout the service. Dog behavioral training is imperative.

Do you send you dog to "Doggy Day Care" in the US at all? If so, how much does it cost? Are they readily accessible?

It depends on the area probably, but most people love their pets, some more than their own children!  Places that have nice weather and an outdoor culture tend to be more "dog friendly" I would guess.

As far as spending money on pets, most people are pretty willing to pay whatever vet bill keeps their pet alive if they can afford. Plus food and the occasional treat. Maybe an outfit or two.