I just finished getting an old TA05 that my cousin gave me up and running (the car hasn't been used since '08).
I'm a complete noob to most RC stuff so I just wanna make sure that the gearing combo I have installed right now won't risk overheating or killing the motor.
See below for the part specs.

• Motor: Trinity Cobalt Pro Tweaked (Part# TRI2002)
  • RPM: 19,178@ 5.0v
  • 27T Brushed
• Gears:
  • Spur: 70T
  • Pinion: 22T
• Battery:
  • 7.2v 3600MaH 6-cell NiHM
• ESC:
  • Tamiya TEU-101BK

Secondary question, anyone have recommendations for cheap brushless ESC and motor. Thinking about changing out the old electronics when I have the chance so I can use some 3s LiPOs I got kicking around.

with that motor you probably want to gear at around 6.2. you will be safe with what you have. (from memory the internal ratio is 2.25)

Agreed. Good place to start.