tuskastus
For the coming summer I've promised to get the two oldest (now 18 and 15) plane tickets to Madrid, Interail (Eurorail) passes, and flight back to US from Helsinki 3 or 4 week later. They'll figure it out, have fun and hopefully don't do too much dumb stuff. Way to bond before the oldest one leaves to college. The middle one gets to do similar trip again in 3 years when he graduates with his younger brother; who in turn will have his trip one year later (hopefully his oldest brother will be available to join him so that everyone will end up with 2 trips with each of their brothers).
I've said that it doesn't matter if they remember it. It's much more important that they experience it.
You can only experience Rome as a four-year old when you are four years old, there is no chance of getting that experience ever again their lives. They can experience Rome again, and again as an adult if they want, but being 4, 8, 12 or teenager at new place is not something that anyone has a chance of ever repeating.
They remember somethings, forget others, but come away with an interest in people, places, environments, etc.