1) The speedometer does not work. Needle not moving. Previous owner installed because it only has 2800 miles on the odometer.
2) Odometer sometimes moves.

Might be ruling out a bad speedo cable because the odometer sometimes works. Any suggestions on what might be wrong?

Thanks!

 

The driven gear is my guess, the cable is my second guess
Pull the cable out of the trans and inspect the gear (both gears, the drive gear in the trans and the driven gear on the cable)
Rotate it and the cable while watching the speedo
Chances are the speedo moves when you rotate the cable
You can pull those cables out from the top (up and out of the cable housing)
That way you can inspect the cable and lube it up as you reinstall it back into the housing

 

What I'd do
Pull the speedo cable out of the trans, remove the clip and the driven gear
Then
Push the tab and remove the cable from the back of the steering column
Pull the cable down by the brake pedal
Then you can pull the cable out

 

I agree with manicmechanic007 ,most likely the plastic gear in the trans, common failure point.