TrackPad Speed and Mouse Speed - Resets to slow every time

Although according to his comment it didn't help Mattias, the first thing I would try is resetting the NVRAM.

Since you don't specify what model your Mac is, I've included two different sets of instructions. Choose the one that applies.

Resetting the PRAM/NVRAM on ALL MODELS EXCEPT for Late-2016 MacBook Pro models

1. Shut down your machine. Yes, a full shut down, not just logging out.
2. Press the power button and then press the commandoptionpr keys. You have to make sure you press these keys before the gray screen appears or it won’t work.
3. Hold those keys down until your Mac reboots again and you here the startup chime.
4. Let go of the keys and let your Mac reboot normally.

Note: When you log back in you may need to readjust some of your system preferences (e.g. speaker volume, screen resolution, startup disk selection, time zone information, etc).

Resetting the NVRAM on Late-2016 MacBook Pro models

To reset the NVRAM on all Late-2016 MacBook Pro models (with and without Touch Bar) follow these steps:

1. Shut down your machine. Yes, a full shut down, not just logging out.
2. Press the power button and then immediately press the commandoptionpr keys.
3. Keep these keys pressed down for at least 20 seconds!
4. Let go of the keys and let your Mac reboot normally.

Note 1: When you log back in you may need to readjust some of your system preferences (e.g. speaker volume, screen resolution, startup disk selection, time zone information, etc).

Note 2: These steps will also work to reset the NVRAM (or PRAM) for all Macs. However, the late-2016 MBP models actually required the commandoptionpr keys to be held down for 20 seconds, while earlier Macs did not require this, but doing so does not cause any problems either.

I called up apple on this. Three things we tried.

1. Login in safe mode, change and save the track speed setting - works.

2. Create another user, change and save the track speed setting - works.

But when I login with my regular account, it does not save the preferred track speed, and resets to slow.

3. The last option we tried and worked was to re-install the OS, using Command + R at startup. This option took 30 minutes or so. But, kept all my data/preferences intact.

HTH.

I haven't tried the solution by @Monomeeth.