Do power plants supply their own electricity?

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)

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.

Wikkipedia - Power Station Black Start

Hydropower Plants as Black Start Resources

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.