When Powers was shot down, the US started with the cover story that it was a weather research aircraft. Then Khrushchev was able to embarrass the US by revealing that they had Powers, evidence that it was a surveillance plane, and demonstrate that the US had been lying to the public.

So, why did the US government lie to it's own people? Why wasn't in their best interest to talk about it publicly?

There's no such thing as telling your own people. Once you inform your own population about it, the enemy knows about it as well. The same as is true with regards secret service such as CIA and NSA. As you say, that doesn't fly here, since the Soviet government knew, and the US government knew the Soviet government knew.

Why you don't want to talk about that could be because its an elephant in a room which could cause escalation because of public discussion. That escalation could be internal as well as external.

Another reason could be that it shows that the US government knew a lot about the Soviets which due to this increased knowledge decreases fear for the Soviets. Which wasn't in the interest of the US government; it was in the interest of both the US and the Soviet government to see each other as the enemy.

A third reason could be, in addendum to #2, that the leadership saw no benefit to it, in terms of propaganda. It might seem the same as #2 but its less nefarious or ill intent.

We may never know the truth about this. Could FOIA requests shed some light?

That's the reason in my opinion.

(A) If the Soviets publicize US penetrations of their airspace, they look militarily weak.

(B) If the US publicizes their penetrations, then (A) as the Soviet public possibly finds out, any fatalities (and there were some) now become part of public debate, and additional parties in the US are now involved in the discussion as to whether or not these should happen.

(C) If either party publicizes their own or the other, then the opposing party is likely to publicize in retaliation. Given that both parties were conducting penetrations to assess air defense systems, this was the PR equivalent of MAD.

Barring a scenario where the president needed a clear "victory" to buttress public opinion, there was no win to be had in the public being informed.

Add in the fact that deep in the Cold War most of the media outlets were still self-censoring "for the country" and there was no one with incentive to disclose.

Though I don't think your (B)(A) regarding the Soviet public was that important. Rather, if the Soviet public were influential, would there have been the gulag system, which ended less than a decade before Powers was shot down?

But the US response to the Soviets shooting down Powers was to lie to the public. Surely the US could have remained silent or non-committal, rather than construct a cover-up, and walk into Khrushchev's publicity trap. According to Wikipedia, Eisenhower said to his secretary "I would like to resign."

That leads to the obvious question - how often does the US government lie to the public?

Lie I don't know, but omit/diverge all the time.