The 3V relay needs 120mA to turn on well. A 2n2222 could handle the current but needs about 5mA base current to saturate it on. So a bias resistor of say 470 Ohm.
MOSFET case you just need to select a MOSFET like AO3400 that has low threshold and turns on well with 3.3V logic. The gate resistor to ground just ensures that relay is off by default even before pin is set to output low.