In the standard web client, search for the hash tag (including the initial `#`) in the "Search or paste URL" form in the top-left corner. Hash tags matching your search will appear just below that form. Click on one you want to follow. A column of toots with that hash tag will appear. At the top of the column on the right-hand side is a small person icon with a "+" in it. Click on that, and you will be following that tag.

Seems like there ought to be a simpler way. I just kind of stumbled into this approach.