I've seen there are a few old threads that I've scoured through here but I can't seem to get it working.
Basically, I have a relay connected to d0, and whenever the power is on it triggers - I've tried a 10k ohm resistor between ground the relay and ground but this just prevents it from working altogether.
I've also put the resistor between the 3.3v and also between the signal (D0) to no avail, the relay just doesn't work at all if a resistor is introduced.
Welcome to the Forum!
Basically you have connected the relay to the port pin used to communicate with your PC / laptop / whatever so unless you do not ever want to download new or improved sketches you might want to consider using a different digital port, not 0 or 1.
You might also want to consider reading the Sticky found at the top of each Forum for information on what you should provide to the Forum when asking for help. Things like your sketch, a diagram of how things are hooked up, possibly even a photo that shows clearly all the connections. The great people on this Forum can only work with what you provide. Provide very little and that is the amount of help we can give you. Provide a lot and things will work out much better and faster.
How; usually connecting a relay directly to an Arduino will blow up the Arduino. Are you using a relay board or are you using e.g. optocouplers or transistors to 'isolate'?
Without seeing code and a schematic (scan / photo of handdrawn one will do), it is guess work. But maybe below solves the problem.
Could you post your circuit with link to relay or relay module? Powering or controlling it with 3.3V could be an issue depending on what you have and how its connected.