Question: I am interested in renaming my characters and worlds. Is that possible? 

I have looked through the UI and could not find a way.

Answer: 
It is possible, but it requires external tools to do so. There are [various inventory editors](http://terraria.gamepedia.com/Inventory_editors) that let you change your player's name and appearance. And, of course, there are tools like [TEdit](http://www.terrariaonline.com/threads/tedit-terraria-map-editor.35394/) that let you edit a world including the name.

Changing the profile/world filename will not rename the world and those files are not in plain text so normal text editor will not do.
[Terrasavr](http://yal.cc/r/terrasavr/) allows you to upload a character save file, edit it, and download it again. It has a wide range of features, and does not require installing any external tools.

You can find your character in Documents/My Games/Terraria.

It does not support world files.