Dogs travel between the UK and Europe all the time these days and there are fewer cases of rabies now than 30 years ago. However, mainland Europe is still not considered rabies free (like the UK and the USA are), so to leave the UK and come back, all dogs need to meet certain requirements.
To travel around the EU, all dogs require a pet passport. The same rules apply to dogs as they do for people – you’re free to travel around most of the EU as much as you want without border controls. Once you have the pet passport, which records veterinary vaccinations and treatments, you’re good to go.