I've had the best experiences meeting people during the actual travel portion of a trip: time on the train, time on a bus, on the plane--anytime where you're basically not sightseeing, but sort of stuck next to people for an extended period. I like to travel with some state quarters, which are really fun to pull out when I'm sitting next to a child. I had a great conversation with a little boy and his grandmother once on the GNER from Edinburgh to London. Turns out, she traveled all over the US when she was younger, and was really excited to see the Mississippi quarter, which was one of her favorite areas.
On another occasion, different trip, same train line, I was headed back to London for a quick one-day stay before moving on. I shared a table seat with a woman who was working on a novel. As her book was set in the US, she asked me questions about Americanisms (like calling 911 in case of an emergency), and she mapped out the most amazing London-in-one-Sunday plan for me that worked more beautifully than she could have imagined.
And I still e-mail a British guy my husband and I met while crossing from Thailand into Cambodia.