What is the simplest meteorological model to operate?
I think you are asking a question with a variety of different constraints. I'll tackle a couple of them.


  What is the simplest atmospheric model to operate?


That would be the Zero-dimensional energy balance model. It has almost zero resolution and no temporal capacity.


  What is an atmospheric model that can be easily installed and run?


The Weather Research and Forecasting (WRF) model is a very flexible, limited area model. This model has successfully been run on what amounts to a child's toy.


  What is a global atmospheric model that can be run for a long period of time, but can easily be run?


While I have never run it myself, it sounds like you may be looking for SPEEDY. I have not heard it is the most numerically stable model, and I cannot attest to how easy it is to run. From the documentation, it seems like it can be run one year at a time, but I was once told that it is typically stable in the range of months or even days.

Atmospheric models are not like a game you can just install and run like MS Word or a game. With almost any atmospheric model, a knowledge of programming in a Linux/Unix framework is paramount.