Game Informer has posted a new interview with Chris Barrett about the places you'll visit while playing Destiny. Go read it!

So does each planet only have one location? For example, is the Cosmodrome in Old Russia the only place we can visit on Earth? Is the old moon base / underground Hive compound the only place we can visit on the Moon?

I'm fairly sure that it was said in the past that there are multiple areas to explore. Around E3 I think Bungie said that the areas that made up Earth combined to make a playable space bigger than Reach, and that the other planets were similar sized. I expect Earth will also feature The Twilight Gap based on the text of the Earth section of Bungie's website. I think Earth may also have a flooded Chicago as well based on past concept and comments.