One of my friends here was impressed with the logistics of bike travel and the logistics of dog travel. However, neither is particularly diificult. In the interest of being informational, here are some tips.
First of all, get a microchip put into your dog. (In the case of pets, this is literally a mark of the beast.) Put a collar and ID tag on it too. Dogs in Europe need something called an EU pet passport. If you are in the EU, get this from your vet. If you are in America or another country, find the website for your country's embassy/consulate for the EU member state in which you intend to arrive. They will have a form which you bring to your vet in your home country a few days before you leave. (Some countries require advanced planning in the form of rabies tests done at least 6 months before arrival. Those countries suck.) Have your vet stamp the form several times with an official looking stamp. Many European countries love official-lloking stamps and will request them whenever possible. The pet travel form is good enough for airlines. Take your dog's pet passport with you whenever you travel.
You need a dog carrier to take a dog on a plane. Your dog is checked baggage, alas. The airline's website will give you information that lets you figure out what size carrier that you need. Put a dog pillow or newspapers down in the carrier. Throw in a toy of some kind, like a kong stuffed with treats. You will also need hamster-style water bottles, maybe a couple of them in large dog-size. Fill one of them the night before and stick it in the freezer. Fill the other at the airport (or before you go to same). My dog had panic attacks for 2 weeks or so after fyling, but was unharmed in the long run. Do not drug your dog.
Most countries want you to buy a ticket for your dog. Alas, as far as I know, you can't get a eurail or interrail pass for your pet. The price of dog tickets varies widly country by country. In Germany, it costs the same as a child ticket. In the Netherlands, your dog can traverse the entire country for around 3€ / day. Small dogs in carriers/bags travel free everywhere. Theoretically, you dog must be muzzled, but so far, this hasn't been enforced in my experience. Heck, my dog is so quiet, so sleepy and blends into shadows so well, that most train conductors / passport agents haven't noticed her at all. I hand them her ticket and they say "This is for a dog!" as if I have tried to save fare by passing myself off as a caninie.
At least in Germany, taking a dog on an overnight train with sleeping benches is insanely expensive. This may be true in other countries as well.
This is a really fun way to travel with a dog, imo. It combines all the best things in life: dogs and bikes. If your dog is small, there exist baskets especially designed for them. There are also specifically designed trailers for any size dog. Doggy Ride is cheaper than kid trailers (that I looked at) and is a better shape for a dog. The trailer hitch will attack to almost any kind of bike. Realisticall, though, your bike should be multi-speed if you want to tow a dog. Three speeds at least. Probably more if you're going through especially hilly areas.
I threw down a very cheap foam mat on the bottom of mine to provide padding. There is also a hook thing that (theoretically) keeps your dog from being able to successfully escape. In practice: make sure all the zippers are closed because your dog can jump out the top. Also, zip the zippers around to the top of the openings rather than leaving them at the bottom, because a dog can "dig" them open by scratching if they're easy to reach. You dog will probably be unhappy the first few times out. I calmed my dog by asking her co-gaurdian to ride behind the trailer saying "good doggie" over and over again, which helped. Now she's happy to jump in, but that took some time. She's still nervous some times, alas.
The trailer folds up to a reasonable size and thus can go on any train.
Obviously, campgrounds are fine with dogs, but most will charge a tiny supplement and demand to see a pet passport. Most two star hotel chains are also fine with dogs, but will also want a supplement of around 5€. Guides published by trourist offices almost always contrain information about whether listed hotels take dogs. Most hotels do. Most hostels do not, but some will is you are with a group that takes up an entire offered room. (ie: two of you in a two-bed room).
Stuff to bring
Aside from the documentation mentioned at the top, bring some sort of small plastic bowl that can be filled with water. Make sure to offer water to your dog very often. And obviously, it will need walks, so bring a leash and also plastic bags in case you walk it in a city. Even Paris is getting serious about making people pick up dog crap, so don't be a part of the problem. Also, getting a ticket sucks. Also bring treats, food and a toy or two. I didn't want to bring a whole sack of dog food, so I pre-measured many days of food into zip-lock bags and brought those. The dog could eat straight out of the bags and I had baggies for picking up after her. My first time out with the pooch, I brouht along her pillow, but it got wet in the rain and was heavy, akward and smelly. Now, I just bring a foam pad, which is lighter. I wanted to get her a thermarest, but they're incredibly expensive in Holland. anyway, she's happy to sleep on the foam.
I prefer taking my dog with me. I can take her into almost every restaurant and even department stores. However, I can't take her into churches, castles, museums or grocery stores. It's ok to tie her up outside a store or a church, but she barks like crazy when left alone in a strange place, which might bother other people. And museums take too much time - I can't leave her barking alone for hours or somebody will think she's abandonned. I also cann't leave her alone in her trailer, as she will destroy it by trying to escape. In the past, I have left her alone in hotel rooms, which was fine, even if against the rules. It's also possible to leave her alone in an airline crate. Dogs find it soothing being locked in boxes and they can't hurt airline carriers.
Is this REALLY a good idea?
Yes! Well, if you like your dog and it's healthy. I wouldn't bring my dog on a grand tour of European capitals because those trips are all about slogging through museum after museum. But if I want to go across a countryside from village to village (which is more fun anyway), the dog is great to have along. Also, people like dogs and it opens up communication.