Arduino Nano problems with ESP8266 this is my first post here because I'm a bit lost at the moment.
I currently have my ESP8266-01 hocked up to the Nano like this: 



The issue I'm having is that it won't even turn on, only when I'm using the 3.3V pin the red led on the ESP8266 turns on. If connected to 3.3V and I connect the CH_PD pin to 3.3V aswell, the blue led flashes 2-3 times and then the ESP8266 turns off.

I hope someone is able to help me
You need to supply power to the ESP to make it turn on.  What you have done at the moment is got the ESP to supply 3.3V to the level shifter, which you need to do.

You need to supply 3.3V to the ESP from somewhere other than the Arduino, it doesn't have the current necessary.