Rear turn signal and brake lights do not work on my motorcycle. How to fix it?

I have a 2005 Honda Shadow Spirit 750. I bought it late last fall and didn't realize until a week or so later that the rear lights do not work. The tailbrake stays on, but does not illuminate when the brake is pressed. The rearrunning lights and rear turning signal lights to do work either. Everything up front works. Anyone have any idea? I think I could fix it myself, but will take it to a mechanic if I have to.

Get a multimeter, learn how to use it and start checking for power and ground. Make sure the battery is good, at least 12.6 volts. You can try replacing the bulbs first.

You need to trace the circuits. Begin by locating the fuses and test them. Then look for the switch for the brake lights and test to see if there's power at the switch. If there's, follow the circuit from there to the actual light fixture.

that the tail light stays on suggests that the light fixture is grounded, so only the hot side of the brake light needs to be tested

Corroded grounds, burned out bulb filaments. Bethcha ain't even looked at nuttin. Take them lights apart with, like, a screwdriver, dude.

If you could fix it yourself, why would you be asking. The only way to troubleshoot wiring is with a circuit tester to check for voltage when it should be present or to check for broken or missing ground connections.

The tail light bulb might be burned out.
The bulb has 2 filaments - 1 for the tail light and 1 for the brake light.
The rear turn signals are not running lights (meaning they don't stay on all the time like the front lights do)

Take it to a qualified motorcycle mechanic and have them fix it.

You can do it yourself - but unless you are mechanically inclined, you might end up getting rear ended when your attempt to fix the brake light fails, and someone doesn't notice you.