How to wire a 240 Volt 500 Watt baseboard heater

I assume you are in the USA. The heater is probably a pure 240V load with no need for the neutral (white) wire. you should simply cap the white wire with a wire nut and tuck it back into the electrical box.

There's not really enough detail to give a solid answer. The answer will depend on the type of thermostat you purchased and what the existing wires are connected to. There's a good chance that you have a switched (red), neutral (white), and hot (black) wire in the wall since baseboard heat often works line voltage. This would be a single pole thermostat with a neutral which I'm not used to seeing (perhaps the old thermostat had any electronics within it). Typically you see 2 or 4 wire thermostats for baseboard heat.

The new thermostat you purchased sounds like a double pole thermostat that supports two hot connections and two switched connections. This would be for a 220v vs 110v baseboard wiring.

You'll need to do some more analysis, the easiest method being to look up the details on wiring your old thermostat and understand what each wire should be. If the white is an unneeded neutral, you'd cap it. And if the black/red you have are hot/switched then those can be wired to one half of the thermostat (only use the black or red wires on the thermostat). Double check the manual on your new thermostat to be sure it supports a single pole baseboard wiring.

If information on the old thermostat isn't available, you can at least verify the hot with a non-contact voltage tester being careful not to electrocute yourself when the breaker is energized for testing. A neutral should have continuity with a known good ground if you have a voltage tester. With the baseboard connected the switched will also show continuity like the neutral. If you somehow reverse the neutral and switched, the breaker will immediately trip when you turn on the thermostat (since you've created a short circuit).

Please don't hesitate to contact an electrician if you can't identify the wiring on the old thermostat and you don't have the equipment to safely test the wiring to verify what each wire does.