Too all those people who simply answer, yes they'd want their children to travel. What if your child is 4? And to those who state that they'd just want to show them how to travel, what if they don't want you a part of their life?
I'm for giving kids choices. But there's a lot to be said for maturity. I'm glad my parents raised me and taught me--we were on welfare for several years, but dang did my mother ever love me. And the problem for me became that she hung on too much, wasn't ready for the last kid to leave the nest (of 6 children) to shirk the marriage-and-kid routine for world travel and moving to a different city, etc. My travel durations grew from 2 hrs, to 2 weeks, to 8 months. Looking back, the point of travel was to full escape the clutches of my parents, siblings, and even families to become who I needed to be. I was conscious that I needed the space and time and experiences to become true to myself, and then return. My mother didn't know it at the time, but it was best for both of us. I love her dearly, and she knows that -- but now she knows little me is independent.
So, what am I getting at? I think that we have a responsibility to raise children the way we honestly think will be best for them, at least until they're eighteen (or until another culturally appropriate rite of passage has been achieved by the child -- for example, in some tribes, a child as young as 14 can choose a wife and make some babies if he builds his own house). I think it is critically important to raise children supportively, vs. laying down the law -- use your parental power from beneath to uphold your children, instead of from above to hold them down. Might seem obvious and easy, but it's probably neither. I certainly can't see holding children down being a good things, because they will either resent you for inhibiting their search for independence, or they will just run away anyway. My mother still doesn't like me riding trains, and tells me that from time to time, but she doesn't nag like she used to because she knows it pushes me away.
So, I'm not sure how soon I would let my kid travel, and whether I would insist on teaching them how to do it even if they didn't want my advice. That's a hard question, because it involves deep, emotional bonds, love. And love is a doozy, because it makes life worth living and can also be destructive when it turns from genuine concern and support to control and selfishness.
I don't know. I certainly want my kids to have the opportunity to travel in any way they choose, riding dirty and thumbin' it or in a Greyhound if they really want to. Travel is a great learning experience. Who doesn't like it in some form? But if they travel to run away from home, I agree with another poster that it would make me wonder where I had gone wrong. They should be traveling to see great new places, not to escape from hell. Even if my kid is in trouble with the law, or is addicted to drugs or is straight up dirty, they are always welcome to come back whenever they want and to phone for help. If I raise a kid, it'll be implicit that I am prepared to provide support.
Oh, and NadayNadie, you might well be the only person on this forum that is not okay with eating out of dumpsters.