Which OS are you using? Assuming it is windows:
1. To minimize it to the system tray: Open the task manager >> Click on options >> Enable the option "Hide when minimized"
2. To enable it on startup: Navigate to your system32 folder (C:/Windows/System32 usually) and look for the file taskmgr.exe. Right click on it, and create a shortcut. Now right click and cut this shortcut, and go to the startup folder (Click on Start >> All programs >> Startup), and paste it there. Right click on this shortcut now placed in the startup folder, click on Properties >> under Shortcut tab - options >> Run minimized. Done.
Now, Task manager will start in minimized every time windows boots. And every time you click on X to close the task manager, it will be minimized to the system tray.