This seems like the most relevant type of feed to be exposed in Chatter REST API, but I can't seem to find any documentation on how to do it.
What I need is the endpoint (if there is one) or the SOQL to get a user's "What I Follow" chatter feed.
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It's in the News Feed resource. The great part about the Chatter API is that it is self-describing; if you go into the workbench, go to the REST Explorer, and start drilling down, you'll eventually find what you want. Specifically, What I Follow is found in:
By using the REST Explorer, you can see other types of feeds you can use: