For those that would prefer a keyboard solution to a VBA/icon one: The
Enter key is not the only key that can be used to enter something into
a cell. The 4 arrow keys can also be used as "Enter" keys.

For example, I have my Enter key set to stay in the current cell upon
enter. For times I need to enter something into a cell and upon enter
have the cell pointer go to the cell to the right, I hit the right
arrow INSTEAD of the Enter key. This enters whatever I've typed into
the current cell and immediately goes to the cell to the right - all
without hitting the Enter key.

Likewise, when I want the cellpointer to go to the cell below upon
enter, I hit the down arrow instead of the enter key. Whatever I've
typed is then entered into the current cell, and the cellpointer
immediately goes to the cell below - again without hitting the Enter
key.

This obviously also works with the up and left arrows as well as any
other movement keys, like Page Up, Page Down, Ctrl+Home, etc. (though
probably less practical).