Picaxe V Arduino.
There is great debate for which is a better platform to use. They both have their pros and cons.
The PICAXE will win hands down for doing simple tasks where speed is not an issue. It is easy to program and there is a range of chips from 8 pin (14,18,20,28) to 40 pin and the built-in interpreter has many functions, you just call in program without the need to understand lines and lines of tricky coding.
Picaxe is an excellent introduction to the world of using micro controllers.
Arduino is a faster chip for more serious and complicated projects, where speed and memory space is needed more. The programming language is harder, if you have no background in programming.
My advice would be to start with the PICAXE and when you out grow their ability, and they can no longer fill your needs of programming ability than move up to the Arduino.
If you are new to programming microcontrollers, then it will take you a few years to outgrow the PICAXE platform. Starting with PICAXE and working up to Arduino will give you a far better understanding of how electronics and coding work.