I'm having a difficult time figuring out the best way to do this. Feel free to explain as if I know absolutely nothing!

In March, I am going to Amsterdam, Berlin, Frankfurt, Beijing and Shanghai. I would like to have a phone in all of these places. Calling and texting isn't really as important as data for things like maps and translation (I speak none of the languages in those places). I have an S3 on Verizon, which has an upgrade available, but the way I understand it, the Verizon S3 is not a global phone for data anyway.

Should I buy a non Verizon unlocked phone on ebay? Will I then be able to buy a SIM card in each of these countries? Any recommendations for a phone that won't cost much but is still usable? Will buying a SIM card be difficult for some reason? Any help is appreciated.