My husband and I went overseas with our cat a few years back.
First, check with your airline for ticketing and procedures. Each airline is different, and some make you pay a pretty good fee. If you don't do this in advance, it might be a surprise.
Most countries require chips. EU has a different chip, so we rented a reader for the various countries to show that they were, indeed, chipped.
Does the country that you are traveling to require any sort of special immunizations? Most of the EU is the same, the passport we got for her was adequate. I know Great Britain has different immunization requirements.
We found a little travel passport for her online that her vet had to complete. Then, we had to go to someplace out by JFK (not at JFK, it was some sort of animal clearinghouse) and they put their stamp on the passport.
They cannot be in their cages/housing units when going through x-ray. Make sure you give yourself plenty of time, as my kitty ran away in Heidelberg. It was humorous, of course, to us. All the officers were trying to catch the kitty, but I'm sure it was upsetting to other travelers. The cat was fine, of course.
Also, be sure to make sure that your hotel accepts pets.
A good website to start would be:
www.pettravelstore.com
Our kitty is very well-traveled! She has been all over Europe and the US. One of the funniest stories was waking up in the middle of the night in Venice and seeing her crawling up the fabric-covered walls! No damage was done. People enjoy seeing the animals when you are traveling and I believe it actually adds to a pet's overall well-being.
Good luck to you!
Traveling with Pets