Which Arduino Board should I start with?
My advice:
 - get a model with USB port (some do not have one)
 - the DUE might seem attractive, but you have less libraries available and less examples, I'd recommend it more as follow up
 - you will get frustrated if you do not know how to code, at least a bit, so if you have not programmed before, you might want to spend some time trying simple stuff just with your PC
 - same about electronics: if you do not have previous experience, it might be preferable to play a bit with a breadboard and some components (LEDs, resistors, switches, etc.); this will reduce the likelyhood that you fry your Arduino. Remember to buy some cables MM/MF/FF, you will end up needing all of them, sooner or later. For powering the board, you can find nice power supplies that cn be powered by a USB cable.

I have bought UNOR3 clones at $4-$5 on ebay, so if you can afford it, I'd suggest you buy a couple, that will enable you playing with I2C master slave and similar configurations.