Is there a shortcut for switching between these three modes?
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This can be accomplished by pressing Ctrl+Tab. This will bring up the shown popup, in which you can select either Vertex, Edge, or Face mode. Note that you can use V, E, and F or 1, 2, and 3 to complete the action using shortcuts.
If you prefer to have own shortcuts, you can create custom ones in the Input section of Blenders User Preferences (File > User-Preferences > Input). Open the 3D View section, and add your shortcut to the Mesh section:
You'll need to fill in the path of the operator to call. For the Mesh selection, this will be
view3d.select_component. Once done, define the keystroke you like, and choose a type from the dropdown box which appeared after setting the operator. Note: Even if the dropdown box already says 'Vertex', choose the option again. The box will be greyed out until you do that, and the keystroke won't function. Save User Settings after that, and enjoy your own keymap!
After pressing the CtrlTab, you can use 1, 2 or 3 in place of V, E or F I find it flow better to end the gesture :)
In every menu/popup you can use these key shortcuts, notice underlined letters.
It's 2019 ;)
Blender 2.80 became much more intuitive. The hotkeys for the select mode are now pre-configured.
Number keys 1, 2, 3 switch between vertex, edge, face selection mode.
It's 2018 so let me add the shortcut command name for Blender 2.79:
mesh.select_mode is what you need for toggling vertex/edge/face in edit mode.
You want to add it under
User Preference -> Input -> 3D View -> Mesh as described by this answer