You can travel within the states with your dog in a carrier if the dog is under a certain size and remains in the carrier under the seat. This takes a bit of training and rules vary per airline and reservations must be made in advance.
You will not be able to bring a dog into Europe in the cabin and if you want to bring your dog at all to Europe, you must start the rather involved Pet Passport process at least 7 months in advance. The dog MUST go in the hold and it is extremely costly. Having just brought a dog from California to the UK then on to Ireland where I live, I can tell you the cost of a one way ticket was more than the round trip ticket for me, and the processing fee on arrival in the UK (you have to pay the cost of clearing the dog and all the paperwork checks) was close to $500 on current exchange rates. So you'd not really want to be travelling with your dog to Europe, probably, unless you have a LOT of spare cash (I am talking the region of $1500 once you pay for all the preliminary tests and vet checks and USDA and vet paperwork,plus the one way ticket, plus the processing costs).
If you google Pet Passport you will get details of the programme (should be on the USDA site) plus you must comply with the requirements of the entry country. Without the Passport, the dog must go into mandatory quarantine for 6 months at your expense. If the paperwork isn't correct, the dog goes into quarantine. So it is a pretty involved and detailed and costly process.
Personally, having flown dogs myself, I think it is far more stressful for a dog to accompany people on long haul holidays like this than it is to be back home, safe in kennels (short flights for a dog comfortable in a carrier would be less stressful for *some* dogs but *more* stressful than the hold for many others! I know that crate trained dogs are likely a lot happier in their crate in the dark hold that squished under a seat amongst noise and smells of people and food). While we may enjoy staying in new cities and finding hotels (easy to do in continental Europe) that take dogs, from the dog's point of view this would be a continuously stressful experience of being taken to strange places every day, left alone in strange rooms each day, crated as you can't leave the dog loose in the room (if your dog barks, this will create problems for you as well), and you as a tourist would be very limited in what you can do with a dog (for example, you can go to many restaurants, but very little else beyond walking around). The dog often needs extra injections because there are different diseases in many areas (the mediterranean for example). You would need to re-microchip your dog because US microchips for some stupid reason do not comply with international standards that the rest of the world uses, so cannot be read unless a shelter or vet has a US scanner, which vastly lowers the chances of your dog being returned if it strays or gets lost somehow.
Europeans on the continent occasionally travel with their dogs but in my experience this is actually fairly rare except in summer, when they are staying put in one place and not moving around all the time -- or when they are very wealthy. Their dogs tend to be well socialised to public places like restaurants and to being crated in hotels. Most American dogs are not, simply because they do not have this opportunity. Hence I think most US travellers on a normal type of vacation to Europe where they want to travel, take trains, visit cultural sites and dine out, would find a dog to be a real liability pretty quickly.
So I'd always weigh up whether people are bringing dogs on holidays more for them or whether the dog would actually enjoy it. In most cases, most dog trainers will agree that dogs are very stressed by travel and are much safer and happier back home waiting to greet their owners on their return.
I have travelled with my dogs, flown some of them twice, taken them by train a few times, and wouldn't do this by choice (the train was OK but clearly stressful as well). I would take them by car to a place where I would be stationary and the holiday involved daily hikes and walks that could involve the dogs (eg it would be a dog *focused* holiday, not a people holiday where the dog is an add-on).
