The concept of "graduation" has changed a lot over time and it's now almost completely retired. Early on, a site graduated when it met the requirements of Beta - in particular, consistently having 10 questions asked per day along with a good answered question percentage. This came with a handful of bonuses and privilege changes - most notably elections for moderators (rather than staff-appointed ones) a beautiful, custom site design and top user swag.
Later when it was clear that there were many sites in need of graduation and not enough designer hours to meet the design needs, "design-independent graduation" became the standard. Sites would leave "beta" based on the same rules as graduation but not get a design, instead being placed in a queue to wait for one. They also wouldn't have increased privilege levels, which were delayed until the site design rollout. The big event for graduation at this point was the first site election.
Most recently, it was announced that dozens of sites were going to leave beta without the concept of graduation any more. By this point the biggest difference between the beta and non-beta state was just the word "beta" on the site banner as elections were standard for all sites, though beta elections are slightly different.
Currently, the work on the new site lifecycle is on hiatus, so more hasn't been released about it and no new sites have left beta in the interim.
As to the process of graduation - there's not really a "staging"... though, I'm not really clear what you mean by that.
When sites graduated with a design, the site was often aware that they were graduating while the design went through a couple of rounds of design feedback. Once those were complete, the site would "graduate" and swag based on the design would be planned and shipped out and an election began. But, both then and now, when a site "leaves beta" (rather than "graduating"), the transition is immediate and often announced at the same time it happens.
Anyway, the sites that are no longer beta are called different things depending on who you ask. To me, they're "full sites" or "sites" (though the latter is ambiguous), to others they're "non-beta" sites... and some still call them "graduated". Some people may also use "designed" to refer to sites with a custom theme (though, two of the designed sites are still in Beta!).
If you're talking about the full lifecycle terminology it's:
Area 51 Proposal -> private Beta -> public Beta -> Full site (or just site)