How to start a terminal window in the center of the screen?

you can use x's geometry to set this.

depending on your x version it can vary.

the man page and it should explain the geometry options

xterm 80x24+nxn or similar.

In gnome-terminal (ubuntu) the geometry specification is slightly different:

gnome-terminal --geometry=114x32+0+0 

As ever the man pages are your friend:

man gnome-terminal

The question title is only about the Terminal, however, it seems the body is more general about "programs start at the top left of [… the] screen".

So, since GNOME v3.30 there is a visible option in GNOME Tweaks, which makes it easy to enable it for all windows:

Just select "Center New Windows" under "Windows".