The trek to Arecibo is a worthy destination for hikers, science nerds and movie buffs alike; it's been featured in Contact, The X-Files, and even a Bond film (GoldenEye). Though its 60-year-old structure has begun to decay, the 1000-ft dish remains the largest curved focusing dish on Earth.
"Monument to Conquerors of Space" is a 360ft (!) statue in Moscow, erected to celebrate achievements in Russia's space program. It sits atop the grand Cosmos pavilion, a museum containing many Soviet-era space artifacts.
Government Building in Merritt Island, FL, United States
Immerse yourself in NASA history at the primary center of human spaceflight since 1968. KSC managed the Apollo, Mercury and Gemini Programs, and was home to 30 years of Space Shuttle launches (1981-2011).
College Science Building in San Pedro de Atacama, Chile
The Atacama Large Millimeter Array (ALMA) is the largest land-based observatory ever built, its 66 massive antennas simulating a telescope 16km in diameter. Situated 5000 feet above sea level on northern Chile's Chajnantor plateau, in one of the least light-polluted places on Earth, ALMA observes some of the first traces of the formation of the universe. Tours are limited and must be registered in advance, but it's a worthy destination for the starry-eyed traveler.
Japan is serious about robots, even the giant fictional variety. Meet RX-78-2 Gundam, an 18m/60ft tall statue outside Gundam's Tokyo flagship store. At night, it lights up and vents smoke, and the wall behind it screens short films from the franchise.