So, normally, when you use your turn signals, you hear the little ticking noise that it is an audible indicator that your blinker is on.

However, on my car, this tic tic noise now comes on and off at random without the blinkers on. Where do I look to diagnose?

Also, the handbrake warning light....does that/should that ever blink (for example, you just got in and started the car but have yet do anything else yet)?

sounds like a worse flasher relay

ticking time bomb?

emergency brake has a flasher in the dashboard when the handle is pulled up.

I suggest you look at the nasty dirty contact at the base of the handle and make sure the brake handle is down all the way.