I'm trying to get my 6V DC brushless motor to push some tray on a track. I connect it directly to some power supply [...] What's the reason for this? Do I need to build a circuit to get motor to stay at a fixed voltage, or is the motor too weak?
You need gear reduction to get more torque. What you have now is rotational speed with less "muscles" - with gearing you get the speed down but it will be able to move a higher load.
So, no circuit, just gearing.
BTW. I assume you by "motor direct to power supply" mean "motor controller directly to power supply"?
A brushless motor won't run without a controller (but it may be built in).
Brushless motors are very fast, compared to brushed motors. The gear reduction needs to convert perhaps 10000 to 20000 RPM at a low torque, to some usable speed at a much higher torque.
Do you have any data on the motor, or a link to where you got it?
Also, the motor uses 6v and 0.50A draw. Without the resistor and a load the motor will actually start moving at 0.1V because the amps raise fast. I'm guessing I need to add a resistor, but how can I calculate and get 0.50A? With 6V, to get 0.50A, it means 12 ohm resistor is needed. However they seem to burn up at 6V. Thanks.
No resistor needed. A motor drawing 0.5A at 6V is the equivalent of a 12 Ohm resistor, but with a 6V supply you don't need a resistor.
The reason a small (1/4W) 12 Ohm resistor burns is that it will see 3W when connected to a 6V source.