Install and play games on Steam solely from an External Drive?

When you come to install Steam, you should get a choice of where to install it, under custom installation if I remember correctly. If not, steam isn't massive anyway.

When you come to install games, you can choose a Steam library to install to. The default is the %\steamapps folder within the steam installation directory. You can choose to use a different folder with each game:

Note: I already had a steam library on my HDD but it was straight forward to create a new one.