The new site is getting hit several thousands of times a day via Google and Bing, but still very few visitors.
Do you just meant that only the bots are visiting or do you mean that people are finding your site from the SERPS and promptly hitting the back button?
There is no "one size fits all" solution for promoting a web site. My own view is to stop thinking of the site as an entity in its own right, think of the underlying business proposition. How is information disseminated within your subject area? That will vary by topic. You may want to post on some relevant forums, you may want to take ads in specialist print magazines. If there is an underlying b&m business then add site details to your classic channels.
For the site itself check how key searches are coming up in the SERPS, if they are not ranking (given sufficient time for the site to establish of course) then look at the content and see if you have the right words on the right pages. If your site is about a rock band then the home page should say that it is for a band playing rock music. If its for a football club then it should say "football", and preferably which of the many versions of that game it plays.