I have become quite connected in society, and an acquaintance of mine is suggesting I present myself at the Shuttered Palace. How do I go about doing so?
The most reliable way to gain access to The Shuttered Palace is through the "Open a way to other parts of the City" storylet that you can find by visiting your lodgings:
To unlock this option, you need 400 bottles of Greyfields 1882, and 1000 bottles of Greyfields 1879.
A less reliable method is to wait for a "The Ways of the Shuttered Palace" Opportunity card to appear, which will provide you with a few options:
This card appears for those with a Persuasive skill of 40 to 80, and offers a "day-trip" to The Shuttered Palace for 50 Silk Scrap and permanent access if you are willing to spend 5 Fate (or Nex, as the case may be).