Opening to the page where I can start to enter a new score.

You probably not using the program Musescore! There is no login page, you don't need to login ...

"What am I doing wrong?"

As explained by HildeK, you are probably not running MuseScore Studio on your own computer.

1. If you have not yet downloaded MuseScore Studio (for Windows, MacOS or Linux only), you can find the free downloads here: https://musescore.org/en/download
2. If this is a first-time download, install MuseScore Studio.
3. Run MuseScore Studio, then menu choice File > New...
4. If you want to create a new score every time you run MuseScore Studio, you can set this option in the menu path:
   Edit > Preferences > General > Program start
   

You wrote:
I click on "Scores" but do not get a drop-down menu saying "new".

Follow steps 1 & 2:

Then click on "New score" as shown above.
Alternatively, you can always use menu item: File > New... to get to the new score wizard.

(BTW: Selecting "My online scores", requires log in to view online scores.)