I imagine CPI numbers are fairly legit. The formula to calculate it is public so a journalist with some hustle could get the data and plug it in to check. Plus, the Federal Reserve does its own inflation report and it usually tracks with the BLS numbers. It would be pretty hard to stack the Fed with yes men like a president could with the Labor Department.
Unemployment numbers on the other hand, oh boy. I would trust the used car salesman at one of those finance anyone lots before I would trust unemployment numbers.