Need to turn on a relay for half a second when constant power is applied with a switch
You're soooo close. But, there are a couple problems with your schematic.  First off, when the trigger pushbutton is not depressed the 555 trigger pin is OPEN. That means it could/will pick up noise and cause false triggering.  Never leave inputs open unless expicitly allowed in the data sheet. Second, the 555 datasheet says that the circuit will trigger when the trigger pin is brought below 1/3 VCC so you need to use a negative trigger, not a positive one like you have. I urge you to read the datasheet carefully. The applications section usually provides an example circuit which is a good starting point.

I also note that there are no protection diodes on the coil drivers. These prevent the inductive kick, when the coil turns off, from blowing up the 555 output stage.

Here are my recommendations for cleaning things up. I'll call the 555 as U1:

1) Use a 10K pullup resistor from VCC to U1 pin 2 to keep the 555 stable (untriggered).

2) Connect the switch from pin 2 to GND, not VCC.

3) Connect the 2nd relay coil from VCC to the switch. Closing the switch will energize the coil but allows you to have the switch connected to GND.

4) Add a 1N4148 or similar diode across each relay coil. Anode to lowest voltage, cathode to highest voltage.

5) Consider using a 0.1uf bypass cap at U1 (the 555) from VCC to GND. It's always good design practice to filter the power supply to any IC.