Speedometer goes all the way up to 120 when first initially starting up. . .then it doesn't work at all

Your Vehicle Speed Output Sensor or Sensors' Eelctrical Connections is malfunctioning.
It is very easily accessed, located on the Transmission; and easily removed and installed.
However before replacing:
Check the electrical connection(s). Unplug and check both sides of the connections for corrosion and oxidation (white stuff). Clean, reinstall and recheck operations.
If this does not fix the problem:
Unplug this Sensor and remove it. Clean off the interior end. Reinstall, replug and recheck operations.
If this does not fix the problem:
Remove and Replace.
I took the liberty to look up the Parts, Availablity, and Pricing. (NO! I do not work for any of these stores/companies)
This is the OUTPUT SENSOR.
Advance Auto Part #S8201 (for Automatic Transmission) for around $17.
O'reilly's Part #S8201 (for Automatic Transmission) for around $18.
Autozone did not have an Output sensor listed, only Input for automatic transmissions.
1) Look the Part up on their web sites to gain insight on what the Part will look like when looking for on the Transmission.
2) Take the Part out and take it with you when purchasing the Part. Compare the original Part with the one you are purchasing to make sure you are getting the right Part.

Let me know if this helped, or if you have additional information or questions. Feel free to contact me at FixYa.com!