You might get away with powering a PIR sensor from a logic line. Check the specs. If it draws less ≤20mA you'd be fine. If you nee to drive more current than that you can use a CMOS TTL Driver or a properly selected N-channel MOSFET. Adafruit sells these, for example.
A relay is another matter. If you this is one of those optically isolated relays and all you're driving is the optioisolator then you might get away with that too if you only need ≤20mA. If you're looking to actually power the relay coils with your Arduino, forget it. it's a bad idea to even have the relay driven from the 5V line on the Arduino. Relays take quite a bit of current, and their load is quite "dirty" and likely to cause problems.
As to "which resistor I might require for the motion sensor & relay", I'm not sure what you mean. You shouldn't need any resistors. If the relay draws more current than the Arduino can supply, a current limiting resistor might protect the Arduino from damage but the relay won't work correctly.
What resistors are you referring to?