There are many energy efficient solutions available but since you have mentioned that you are a noob, I'll try to provide an easy solution which won't be energy efficient but should work fine for your project.
Step 1: Get a 220V to 12V wall adapter. Choose a current rating to suit your requirements. Most of the current will go in DC motors unless you have a long LED strip tied to your project.
Step 2: Decide whether you are going to use an off the shelf arduino or you are going to make one yourself. If it's a ready made material, then there is a DC jack where you can plug in 12V directly to power up the arduino. Problem solved. If you are rolling your own arduino using atmega IC, then use 7805 IC to get the power required to run your micro controller. Usually arduino consumes very low current and thus you should be fine with a linear power supply without heat sink. But if it's really going to consume 500 mA, better add a suitable heat sink to the 7805 IC.
Step 3: Most likely the LED should work with 5V or 12V. You might have to change the series resistor. Till date I haven't seen an LED which works on 3.3V but not from 5V or 12V. (Please provide the link.) I'd suggest you to wire up the LEDs on 12V line with suitable resistors and use a transistor to turn them ON using arduino GPIO pin.
However if you are sure that you can't make it work using 3.3V, then you can use a similar solution as in step 2. Get a linear regulator for 3.3V and power the LEDs using its output. you can use LM1117-3.3V for this purpose. And yes, a heat sink too.
Here is another answer I provided to a similar question. This requires a little knowledge to implement but you should know the better ways of doing things.