How do I manage the two grey buttons?

Those two grey buttons are known as Calls To Action.

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You can use them to invite your readers to take advantage of special offers, visit a certain page, subscribe to your blog or whatever you like!

To edit them, navigate to Dashboard >  University Theme > Featured Area

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Scroll downwards and you will find this:

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The Button Text tells your reader what to do, and the Button URL takes them to where they can do it on or off your blog. For best practices, try to keep these links pointed back to a landing page on your blog where they can take advantage of your Call To Action.

Edit, save and review.