Phone not updating Market to Google Play?
From the official announcement on the Android Developers blog (emphasis mine):


  We’ll be rolling out Google Play to devices in a phased OTA update, starting today and continuing over the days to come. With the update, the Android Market app will upgrade to the Play Store app and the Music, Videos, and Books apps will upgrade to Play Music, Play Movies, and Play Books. This update is for devices running Android 2.2 or higher, and users on other devices will continue to have the same access to your apps as before. 


Just give it some time to work. For what it's worth, I got the upgrade almost immediately after I read the blog, and the new Google Play appeared in the apps instead of the old Market. It's pretty much the same as the previous Market app, bar the new branding. Also, check the version of your market app -- Google Play is version 3.4.6 or later, the old Market is 3.4.4.