Keep your airlines in the loop.
First of all, not all airlines will allow your dog to travel with them. But there are a lot who will. When I flew from Portland to Amsterdam with Olive, we flew Delta. Delta was very accommodating and just asked that I either provide an emotional support animal note and tell them ahead of time, or if I didn't have said note, that I specify the exact dimensions of the carrier she would be in. In my case I didn't use a carrier because it is not required if your dog is an emotional support animal (hint hint - visit your doctor). Contact the airline directly before booking because even if the airline typically allows animals, sometimes there are certain flights that have stricter rules. Also better safe than sorry, check in with your airlines again a week or so before travel to make sure they're aware of your furry addition.
Choose trains over buses whenever possible.
First of all, most of the time dogs aren't even allowed on buses. I found that out the hard way after getting turned away from my connecting bus to Prague so if you're trying to go as cheap as possible - transportation isn't one of those areas you'll be able to get many bargains on. You will most of the time also have to pay for a ticket at half fare for your pet (at times I even paid a full-fare ticket).
Trains in Europe:
If you can, take trains that don't require reservations as much as possible. Look for ICE/IC. If you don't need a reservation then you don't have an assigned seat number which means you can find a comfortable spot for you and you pup and not be confined to one area (where you'll most likely have neighbors). I've found they're pretty tolerant of your pup sitting in an empty seat as long as you have something for them to sit on like a jacket or blanket that covers the entire area. You don't want to leave a fur-filled area and ruin the leniency of train policies for future travelers.
Be aware that DOGS ARE NOT ALLOWED ON NIGHT TRAINS. Another unfortunate situation I encountered while trying to move long distances. They only allow your during day transport in Europe and this can take 2 days of travel rather than just one night.