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I tried looking it up but there's no answer for this. This is mainly concerning nuclear power plants or natural gas plants, where they would have other buildings attached on site to operate the plants.
When they run the plants does it go to the grid and then back to the facilities? Or are the building circuits directly powered by the plant electricity, then the remaining electricity goes to the grid?
I also assume wind farms and solar PV farms don't have this concern, but I could be wrong.
When a power station is up and running the power it generates can be used to power its internal operation.
However, many power stations do not have the ability to bring themselves back into service after a major outage. (referred to as a "Black Start").
It may not be worth the capital expenditure to provide adequate backup power (that would only be used very occasionally) to perform start-up; instead, they would use power from the power grid relying on other power stations in the same region. At least one would have to have the ability to bootstrap itself.
Nuclear power plants in particular have extremely critical needs for power even when they are not generating, as decay heat requires circulating cooling water through not only the reactor long after shutdown, but also possibly some of the recently spent fuel stored in pools of water outside the reactor.
As a result they are not only able to draw from the grid, but have local diesel backup generators.
One of the fundamental causes of the disaster at Fukushima was that the diesel generators were flooded by the tsunami, leading to loss of power for active cooling.
(The backup generators are sized for emergency cooling only. Very rarely have they been officially sufficient for normal operation in the sense of being able to perform a "Black Start". Some searching will find commentary from plant engineers who think they could get things slowly started if authorized in a dire need by using limited pumping and natural convection, but it would be well outside normal procedures and precautions)
They are powered off the grid. Power coming out of the plant directly is not suitable for end use, it needs to run through transformers and other conditioning/protection equipment. You also wouldn't want a fault in one of those buildings to take down the plant.
Allow me to explain with an example, I will be describing a hydraulic generating station, which is the simplest of all generating stations in terms of complexity of equipment. The station in this example does not have black-start capability.
When they run the plants does [the power that they generate] go to the grid and then back to the facilities? Or are the building circuits directly powered by the plant electricity, then the remaining electricity goes to the grid?
In theory* it doesn't work that way. There isn't one type of connection that supplies power to the grid, and a different type of connection that takes power from it. Whether or not any given piece of equipment produces power or consumes power depends only on the equimpment itself, and not on how or where it is connected. If you set up a solar power system with a grid tied inverter in your home, the inverter connects to the grid through your circuit breaker panel and your existing electric meter. You don't get a separate wire out to the power line that runs past your home.
It might help to picture one of those old factories in which machines were powered by an overhead shaft that ran the length of the building and was turned by a steam engine. Suppose you owned that factory, and you decided that you needed more power. You could easily remove a machine that was taking power from the shaft, and put a new engine in its place to add more power to the shaft. No need to change anything else. The shaft doesn't know or care which direction the power flows. Neither does the electrical grid.
* I can't speak about things they might do for safety or regulatory reasons.
During the northeast blackout of 1965, Ravenswood No. 3 in Long Island City, Queens, NY, also known as Big Allis, a 1000 MW steam turbine generating plant, had 14 of its 15 turbine bearings damaged by lack of lubrication. The lube oil pumps were powered by the grid, so when the grid failed, so did the bearings.
