Can move the gearshift but car actually stays in neutral and wont go anywhere?

Hello, this sounds like a problem with internal transmission engagement. This very likely is mechanical, but can also be electrical. A diagnostic tool can be used to monitor transmission shift lever selection via the transmission range sensor reading, and can then monitor shift solenoid positions and intended gear engagement. If no engagement occurs while all this information is accurate, you likely have an internal mechanical problem that could need partial tear down, or at least dropping the transmission oil pan to inspect. First, though, check to see if the fluid for the transmission is clean and full.

This is a very common problem with the Taurus transmissions .. The splines inside the torque converter have stripped out and no longer driving the fluid pump - resulting in NO engagement in any gear.. To confirm this - simply check transmission fluid level with engine off - mark or note fluid level on the dipstick .. Start engine and re-check transmission fluid level again - it should now show less fluid that it did before with the engine off .. IF no change in fluid level - transmission got to be removed for repairs. It is that simple!