How to show user first name + last name in top bar in sharepoint 2013

That part is coming from a User Control named welcome.ascx. There are multiple ways of changing, most easy (but not necessarily the most recommended - because changing out-of-the-box controls should be avoided) is to edit the control in the CONTROLTEMPLATES folder in the 14/15 Hive.

Another one would be to build your own control by inheriting from he original one. Best reference on Wictor Willen's blog http://www.wictorwilen.se/post/having-fun-with-the-sharepoint-welcomeascx-control.aspx