Do I have to download anything on the Downloads page to use ForeFlight Mobile?

When your device is connected to the Internet, ForeFlight Mobile will automatically grab any data it needs regardless of whether you have set up any items to download. 

You should download those items, such as charts or terminal procedures, that you will need while not connected to the Internet (i.e. in flight). This will ensure that you have what you need in flight and will also allow you to view items within ForeFlight faster.