So, I'm a complete arduino noob and I'm thinking of starting my first project. I want the arduino to control a simple led strip, like this

I know the PWM outputs of the arduino can`t stand much current, but my question is, does the strip get powered by the +12V contact and the RGB ones are only for PWM? Can I directly connect my arduino to the RGB contacts to control it?
Your Arduino won't be able to handle more as 5 volt and 40mA per pin.
But you can control NPN-transistors or N-channel mosfets using your arduino, which can handle a much higher voltage/current. Which type of transistor may depend on the length/ nr of leds on your strip.
This instructable shows you how it's done.
Apparently the writer only used a 12 volt power-supply, which created the need for a 5volt regulator for the arduino as well, but you won't need it if you already have a 12 and 5 volt supply.