There are a few examples of international travel.
During the Triwizard Tournament:
- The Beauxbatons arrived in a large carriage pulled by Abraxan horses.
- The Durmstrang arrived via a large ship which could travel underwater.
In those two cases, they were bringing students who likely hadn't all learned to Apparate yet.
When the Weasleys were going on vacation to Egypt, there is no mention of how they all went. I think the assumption is either Portkey or perhaps an international Floo Network, given the complete lack of competence the Weasleys demonstrate towards Muggle methods.
Later Voldemort traveled to and from Nurmengard (Germany or Bulgaria) back to England. He flies until he is close enough to his destination to Apparate.
Since you could in theory fly anywhere, as long as you were patient enough, you could use a flying car or motorcycle, a broomstick, or any creature which can fly and support a human (Thestral, Dragon, Hippogriff).
In Fantastic Beasts and Where to Find Them, Newt both arrives and leaves from New York via a traditional steam ship. Now, it isn't clear whether that was by necessity or not. He kind of sneaks into America with a suitcase full of illegal beasts. Perhaps if he had gone via Floo Network / Portkey, he would have had to go through magic customs or something.
Additionally, On Pottermore, under Ilvermorny School of Witchcraft and Wizardry, Isolt Sayre traveled from Ireland to America via the Mayflower. She was also traveling incognito, and likely didn't have the normal magical contacts that would enable her to travel via magical methods. The two boys she rescues (who had magical parents) also traveled by boat and had settled in Plymouth. Presumably this means they were on the Mayflower together, and didn't know it. Gormlaith also travels by boat to America in order to seek her revenge against Isolt.
I've written this song to commemorate travel in the Harry Potter world:
All right stop, Apparate and listen
Voldemort's back with a brand new invention
Flying! grab a hold of me tightly
Floo Network's dope like a Train leaving nightly
Will Thestrals stop? Yo, I don't know
Kill the lights and I'll Lumos Maximo
To the extreme I rock a Wand like a vandal
Find a Portkey and grab a Broom by the handle
Atop a big Hippogriff
Find a horcrux near a seaside cliff
Deadly, like Avada Kedevra
Azkaban Prison is another matter
Snitch it or Quidditch
You better set Sail
You better not hex or you'll be Dragon tail
If there was a goblin
Yo, I'll be slick
Transfigure a teapot with a simple wrist flick
Harry Harry Potter