Common Wire sizes for connecting peripherals to arduino
Depends what you’re doing.

For low power (&lt; 12v, 500mA) peripherals and digital signals:

If you’re going to use breadboard (and you really should start with breadboard) you’ll need solid core hook up wire. You can find pre-cut, pre-stripped sets with a variety of lengths and colours if you’re looking for neatness and simplicity. 
(An example can be found at: http://www.maplin.co.uk/p/jump-wire-kit-fs65v)
Usually it’s 22AWG wire, so you could get a reel and cut it yourself if you’re going to need a lot or are going to solder it in place.

After you’re done breadboarding and want to make something more permanent you should look at rainbow cable. You can tear it to get anything from 2 to 32 way cables, and it’s almost impossible to get the wrong way around. You can solder it directly to a PCB or crimp connectors onto the ends for more flexibility.
(You can find this stuff everywhere, but Sparkfun is always there if you can’t: https://www.sparkfun.com/products/10649)

For higher power (> 12v, 1A): 

Keep it away from the Arduino! Use switching transistors, relays, or MOSFETs and use cabling that meets, or preferably exceeds, the power that you’re going to put through it.

I don’t know where you’re going to buy this, but in the UK I use Mouser, Farnell, and RS for most of my electronics supplies. Never buy cabling off Ebay, a lot of it is counterfeit and won’t meet the claimed spec.

Hope this helps.