Ok that circuit is not one for a beginner.
Simply get an SSR (Solid State Relay).
The simplest ones just switch. You can get ones that will dim, they are sometimes called phase controlled SSRs. And they are a simple low voltage connection to the Arduino. You then use PWM to control the brightness of the opto coupler LED that is built in inside them.
This is an example of one but I have seen them at one third of the price on ebay
http://www.ebay.co.uk/itm/Crydom-10PCV2475-Solid-State-Relay-Proportional-Power-Controller-/251157360536?pt=LH_DefaultDomain_0&hash=item3a7a252f98