How do I get the Smithsonian Channel
You would need to subscribe to the tier that has that channel at the price for that tier. You can't add just one channel.
I think in most areas, it's included in the Preferred tier, which is typically about $18 more a month than Starter. It also includes many other channels, along with the Encore movie channels (Encore is part of Starz and airs movies and series after they air on the main Starz premium channel.)
Smithosonian does make some content available on-line and through apps, but it's very hit-or-miss what is there. No subscription required for on-line access, or even apps for phone, tablet, or streaming devices.