The easiest place to track our progress (and where that screenshot came from) is on the original "Area 51" proposal:
LEGO - Area 51
As of today (11 September 2018), we're showing:
The key metric that we're down on is questions per day - we're fairly slow to generate new content, and this is now the key driver of a site's consideration for graduation (emphasis from the original):
from now on, when a site starts to consistently receive ten new questions every day, we'll consider it for graduation. This is not 100% automated; CMs will still be manually checking on how individual communities are doing, and some sites might still graduate "earlier" or "later" than their question activity alone would suggest; but it's an effective rule of thumb.
It used to be that sites that didn't meet the criteria were closed fairly swiftly, however as you can see from Shog9's answer here - with our high answered rate, and the dedication of our users, we're in a stable, good place.
See also this answer to "How do I know if a beta Stack Exchange site is growing well?":
Graduation needs a body of expert content, and a community to maintain and expand it. It's much harder to quantify that - there are some stats that can be manually generated by the SE developers, such as the number of high-rep users who visited in the last few weeks, and so on. And there are publicly-visible indicators such as the liveliness of the site meta and its chat. As more and more sites come through Area51 to graduation, we're all learning more about what indicators work, and what don't.
2018-10-03 - Edit to add:
The new StackExchange Theme includes the following module in the right bar on the homepage of the main site (and other Beta sites):