I believe that even better thing to do than sharing questions on social media is to try and think about questions that would bring people to this site automatically and in high numbers. I am writing this response primarily as a reaction to a really surprising amount of views I have noticed on this question.
Although being only a couple of hours old, it has already brought 360 people to the site and the number is growing rapidly (~1 view per minute). It is one of the most viewed questions already.
What I believe we should do is to try and think of not only expert questions, but that we should have questions and answers to regularly asked questions people have about vegetarianism. These would serve as an entry point and an introduction to the site for people that would otherwise never hear about it.
While I know that non-expert levels of questions are being frowned at on SE, there are two reasons why I think we should condone these in this case:
- Even for a basic question that people have about vegetarianism/veganism, there can be a great and expert answer and this site should provide such.
- As suggested at Area 51 page, most of the traffic should eventually come from search engines. This will happen either really slowly or not at all if we do not allow for our little SEO optimiztion.
EDIT: I felt it necessary to add that, as Erica mentioned in the comments, my answer was based on a mistaken premise that the extraordinary number of views on that particular question was driven by searches from search engines. It seems it was not and that the traffic was due to it being promoted at the Stack Exchange Hot Network Questions. I believe that makes my answer rather less valid, taking into consideration this answer from Robert that Zanna mentioned in the comments. I am not deleting it though, for future reference.