DO promote on:
Blogs
There are a number of blog posts out there already about the Stack Exchange site and these have been useful for getting us to beta in the first place, such as the following:
https://www.akshaysura.com/2016/09/19/how-can-you-help-the-stack-exchange-beta-site-for-sitecore/
http://blogs.perficient.com/microsoft/2016/10/sitecore-stack-exchange-needs-you/
http://www.flux-digital.com/blog/sitecore-stack-exchange-site-needs-your-help/
Hopefully, blogs can continue to be a source of promotion for the site going forward.
Social media
This is Probably the biggest forms of promotion for the Site. Especially Twitter and Linked In. Perhaps there are other Social Media Channels that can utilised further though.
Sitecore community update mailing list
This mailing list by Akshay Sura has been useful in getting the word out in the Community:
https://www.akshaysura.com/sitecore-community-update-mailing-list/
Sitecore Slack community chat
Slack has been a great way of getting people to engage on the Sitecore Stack Exchange site and ask/answer questions there. Especially for maintaining the answer to help others in future instead of them getting lost in a slack Channel.
Sitecore user groups
Whenever attending one of these or even presenting at one; try and see if you can get a 5-10 minute slot near the end - to just present or even just talk about the community effort going on here. Doesn't need to be any long explanation or complicated series of slides - just something simple. For example:
"Hey, for anyone who doesn't know it yet - the community has taken initiative to get a Sitecore Q&A site going. Its url is http://sitecore.stackexchange.com and you'll find lots of useful information in there already. If you run into any Sitecore-related questions in your daily work, this is an excellent resource".
Among colleagues
Make sure your colleagues are aware that Sitecore Stack Exchange exists and welcomes Sitecore-related questions, as well as good answers.
This site is still new, so the people you work with are likely to be unaware it exists.
DON'T promote on:
Stack Overflow
stackoverflow.com is not an appropriate place to promote Sitecore Stack Exchange. It is against the SO rules to do so:
And please do not solicit content or users by posting comment on Stack Overflow; that is not allowed. There is generally no problem with a little crossover between sites...
community.sitecore.net
The consensus is that community.sitecore.net is not the right place to promote Sitecore Stack Exchange.