Displaying text on a typical HD44780-compatible LCD with an Arduino isn't difficult at all, and with a little extra work you can create custom characters as well. However any form of animation is quite hard to make with letters of the alphabet and so on. However with a little work you can create reasonable animated graphics with these LCDs thanks to the guide by Instructables user "uselessbagofmostlywater".
They run through the process of not only wiring the LCD to an Arduno, but also show you how to create an animation plan into the required code for display with the Arduino. Over time you'll be able to create your own versions and recreate some interesting displays that can push the limit into these LCDs, for example:
If you're using an LCD with your Arduino-based projects - instead of building your own LCD module onto a breadboard, save time and move forward with the Freetronics LCD & Keypad shield which contains a bright 16x2 character LCD and five buttons that can be read from only one analog input pin: