Is the new site at the same domain as the previous one? If so, I'd get a hold of a sitemap from the older version of the site, plus the results of a "site:domain" search to make sure there are proper 301 redirects from legacy pages to new URL structures.

Is the staging site still live & accessible to anyone who has the link? If so, I recommend either shutting it down or password protect it, possibly via htaccess.

Regarding old site to new site mapping, there are likely pages linking to old content that are now resulting in 404 errors. That's going to be where most of the "juice" is. Take a look at the results in Open Site Explorer, as well as Google Search Console's links to your site section, and see where your current backlinks are pointing. If there are equivalents on the new site, having a 301 redirect from an old URL to the new equivalent will help guide visitors to the right content. If there's no equivalent, redirecting to the home page works in a pinch.

That's good that the backlinks you can find are to new, existing content. Also check the crawl errors in Search Console to see if there are potential 404's that are getting linked.

With regard to the staging content, theoretically you could assign a 410 status code to that directory, thereby telling crawlers that the content does not exist and won't exist ever again. Search Engine Watch has a quick rehash on Google's approach to 404 and 410 codes.

For the remaining urls from the old site, since they don't have current equivalents, a 301 redirect to the home page would be a good move. At least then if someone has an old link to your site, they're getting to something that isn't a 404.

What filters do you have setup?

Generally, I prefer to set up two views on an Analytics property, one containing all of the data, and one with filters. This way if the filtered data is looking lower than expected, there's still the complete view to compare against. It also makes it easy to test the efficacy of the filters that were created.