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Clearly, there have been some sorts of conspiracies and secrets in the past, and this doesn't mean that we should take the claim the moon landings were faked very seriously. But what if you found out that there were conspiracies centred on you (or your friend Dave)?
Would (should?) that make you suspicious that there could be others?
If by conspiracy, you mean something like "an agreement between two or more people to commit an illegal (or if you prefer for this discussion, illicit) act, along with an intent to achieve the agreement's goal along with at least one overt act to move forward to the goal" (which is a summation of how it is often defined in U.S. legal contexts, though it does vary by jurisdiction), then they clearly exist. People are charged and convicted of illegal conspiracies all the time.
If you somehow believed that conspiracies of that nature did not exist, and then suddenly found out that at least one conspiracy was real, then it would likely shift your thinking (either informally or through Bayesian reasoning) to make you believe there were likely other conspiracies. You would also be correct and they are relatively common.
One famous example, though it predates the definition I summarized above, would be the Guy Fawkes Conspiracy to blow up Parliament. Racketeering almost always involves a conspiracy in the legal sense.
On the other hand, if you specifically mean the informal term of "conspiracy theory" where a supposed shadowy cabal was hiding the truth about something, it might be different. Finding out that the moon landing was in fact faked would not make more more likely to believe that 9/11 was an inside job or vice versa unless there was an explanatory link. Finding out that the moon landing was faked and the same organization that faked it has an affirmative reason to have misled the world about the events on 9/11 would be very different because it provides a relevant connection.
As David Ray Griffin, a process philosopher has often argued in his investigative works on 9/11, conspiracy theories, like any other theories about concrete world, can't be ruled out or affirmed a priori. Conspiracy theories, similarly to other theories about the world are proved or disproved by evidence while observing the principles of the scientific method like avoiding confirmation bias, independent testimony in case of conflict of interest, etc.
So to say that conspiracies are a priori impossible is unwarranted. Viewed a priori, anything compatible with man's nature is possible. We know humans can lie, cheat, be greedy and overly self-interested, and thus behave in malicious ways. These are human tendencies that make conspiracies a priori possible.
Conspiracies in fact may be even said a priori to abound in politics as the force of human political and economic interests become the most strong there. Indeed, consider military coups that have been common in contemporary history. What are they? Well, they are conspiracies. Now to have some other conspiracy theory like about JFK assassination or 9/11, is to propose another instance of a same recurrent category of human behavior that in such case may or may not be true.
Now the suspicion of a conspiracy may be based on paranoia or pessimism or ideological tendencies, but ultimately, regardless of the incentives or impulses, a theory falls and rises, or is weakened or strengthened always by evidence obtained through responsible research, observation and analysis.
That sums up all you need to know about the concept of conspiracy theories.
As others have already pointed out, that depends very much on the nature of the conspiracy. Like 9/11 was undoubtedly a conspiracy... by al Qaeda. Seriously it if you take the Wiki definition of a conspiracy:
A conspiracy, also known as a plot, is a secret plan or agreement between people (called conspirers or conspirators) for an unlawful or harmful purpose, such as murder or treason, especially with political motivation, while keeping their agreement secret from the public or from other people affected by it.
Then what they did was a conspiracy. So conspiracies do exist and yeah if something exists it is likely that it will be taken into consideration when it comes to the assessment of reasons for an event.
The thing is "regular conspiracies" are complicated and have a lot of dynamics that are directly opposed to each other:
Though when it comes to what are colloquially called "conspiracy theories" you kinda get into the domain that is almost religious. In the sense that you believe in an entity that is all powerful, all knowing and almighty (* within limits) and that constantly, successfully interferes with the world on both a global and personal level without leaving any trace or being uncovered. Seriously that could literally serve as a definition of a god.
And that's not even an exaggeration given that some literally believe in aliens, where the "advanced tech = magic"-formula comes into play or stuff like the simulation theory where the creator of the simulation would be some kind of "god" for that universe.
It's also a self-fulfilling prophecy in that if you believe in their absolute power to conceal their actions perfectly, then the absence of proof is proof itself. And I'm not speaking of actual evidence like a used car being perfectly clean as proof of cleaning, but the absence of any evidence being evidence itself. And the more your paranoia takes hold of you and the more you distrust others, the more it becomes impossible for you to verify or falsify any claim as you rely on the expertise and witness of others. Which you can conveniently dismiss as liars in on the conspiracy. Upon which you'd be approaching absolute skepticism up to nihilism and the inability for any sort of epistemology outside of the "fact" that the conspiracy exists.
So sure if you believe in a god that is able to interfere everywhere and at every time and level then that is THE primary variable of your universe that you have to account for and you'll potentially see that everywhere.
If you don't, then you'd realize that for any human actor or actors faking the moon landing is probably more difficult than going to the moon. Like you'd employ 400,000 people in the Apollo project that you'd have to provide with secrecy and/or bullshit jobs that don't talk to their friends and families about their job, that create conditions on earth that aren't possible on earth, that stand the test of time even up until now, that they were confident enough to show to 600,000,000 people, that have since been replicated or confirmed by other nations who at the time had no interest of granting the U.S. that success if they could do that. And the longer it goes the more extraordinary the secrecy measures would need to be, while the more insignificant the achievement would be. Like sure it's a historic feat, but if it didn't happen in 1969 but does happen now what's the point of hiding that knowledge in the first place.
So at some point you'd probably take a leap of fate into the abyss and if you do it's hard to recover.
The other answers have skirted around the crux of the question
what if you found out that there were conspiracies centred on you (or your friend Dave)?
As one answer said, you need an explanatory link between 9/11 and the moon landings to think that one is good reason to question the other. I think if the CIA is plotting against you, then you should be suspicious of them doing so consistently and in more than one way (if you have evidence of one, it seems reasonable to suppose it's not alone). I think day of the tentacle argues for this.
I could be wrong, but if we're talking about secret plots, rather than mere surveillance, then there is an explanatory link - you. Though of course that way madness does indeed lead.
