You can use just a simple Transistor (e.g. BC548) or FET (e.g. 2N7000) interface circuit. See for example the "Output Devices 5 - Solar & DC “Toy” Motors" in PICaxe Manual (section) 3, page 12. Replace the 6 volts rail with a 12 volt supply, put the Relay in place of the Motor, and of course change Pin 1 to Pin C.2. The diode is still required, but most (NPN) Transistors or FETs should be satisfactory. If using a FET, add a high-value pull-down resistor (22k - 1M) between the Gate (or PICaxe Pin C.2) and Source (0 volts line).
There isn't really any more to show; "Standard Circuits 3 and 4" (page 8) are similar, just change the 5v to 12 volts, but only on the Relay; the PICaxe rail must remain at 3 - 5 volts. I can't give any more indication of pin numbers as you haven't named your PICaxe version.
I don't know what you mean by "relay is 30Am and the coil draws 360 mAM", but a coil resistance of 30 ohms and current of 360 mA (milli Amps) seems very high (4 watts at 12 volts). That would need a better transistor, perhaps a BC337 with the 1k resistor reduced to ~220 ohms, but the 2N7000 might be sufficient, or the BCX38C from the manual, etc.. However, the "Output Device 11 - Solenoid & Solenoid Valves" on page 24 may be more appropriate (again changed to 12 v) for that current. What components do you have available?
I have many electronics components. Among other things, I made this go-kart for my daughter incl. PWM motor controller that can handle 50A. (See link)
However, I have not worked so much with picaxe ... hahaha
Yes, your circuit is wrong; it prevents the Relay "seeing" the 12 volts (only the 5 volts from the PICaxe). Both the Source and Drain (or Emitter/Collector of a transistor) can amplify the Current, but only the Drain/Collector can amplify the Voltage. The higher voltage is needed to drive the required current through the resistance of the Relay (Ohm's Law: Current = Volts / Resistance).
But don't forget the Diode, it's important to "protect" the other semiconductors from the relay inductance. Best to connect the diode directly across the relay coil - but make sure it's connected the correct way around (the "bar" end to the positive supply voltage).