Regarding checks and balances the president, Congress, and the judiciary have on the bureaucracy, which ones are most effective in their wide use?

First of all, please note that (in the way we usually use
the term) the president does not have a check and balance relationship with the
bureaucracy.  Checks and balances are supposed to work
between branches of the government.   The bureaucracy is
part of the president's own branch and so the president does not check or balance
it.

The most effective check or balance on the bureaucracy
is the legislative branch's power of oversight and budget.  The legislature has the
power to set the budget for any bureaucratic agency.  It also has the power to summon
officials from the agencies to appear at Congressional hearings to explain their actions
(and often to be chastised by every member of the committee).  These powers are used
quite often as a way for the Congress (particularly when the President is of the other
party) to affect what the bureaucracy can do.