I'm not aware of a way to switch from the Windows screen to the Applications screen, but there are ways to launch applications and switch windows using the keyboard in gnome-shell.

Alt-Tab

Your question only mentions the Activites pane, but I'll mention this method for completeness, and because it most closely matches the behaviour of several other environments: To switch between applications, use Alt+Tab. To switch between multiple windows of the same application, use Alt+`. Note for non-US-keyborad users: To switch between multiple windows of the same application, use Alt+key above Tab.

Searching in the Activities pane

The "type to search" field in the top right of the Activities pane is one of my favourite parts of gnome-shell. To launch an application without touching the mouse, press the "Win key" (here I'll call it Super) and type the first few letters of the application's name. A list of applications matching your text will appear. Use the up and down arrows to highlight the one you want, and press Enter. For example, if I press Super and type "libr", the applications in the LibreOffice suite are listed for me, and I can select one.

The same method applies to switching windows: if you select an application that's already running, the shell will switch you to that application instead of opening a new instance of it. For example, if I already have Firefox open, and it press Super and type "firef" and press Enter, I get one of my existing Firefox windows instead of a new one.

So what if you want a new instance of an application that's already running? Select the application you want and press Ctrl+Enter.

The Window Navigator extension

This is an extension that allows you to select windows in the activities pane using keystrokes instead of mouse clicks.

These instructions explain how to enable a PPA with some gnome-shell extensions. The relevant commands are as follows:

sudo add-apt-repository ppa:ferramroberto/gnome3

sudo apt-get update

sudo apt-get install gnome-shell-extensions-windows-navigator

After installing, restart gnome-shell (Alt+F2, type "r" and press Enter). If you use gnome-tweak-tool, you may need to launch it and enable the Window Navigator extension under Shell Extensions.

Once the extension is installed and enabled, you can hold Alt in the Activities pane to see a number hover over each window. Press a number key to switch to its corresponding window.

Update

There are plenty of other tips and explanations in the Gnome Project's GnomeShell Cheat Sheet.