the first moon landing had tons of conspiracy theories revolving around it before they even landed, and after they did it just took off. (but dont all conspiracies)
Some of the major things people where going on where how the pictures taken from the lunar landing site had no stars in the sky you would think their be tons with no atmosphere. skeptics say because astronomers would be able to tell a fake. the reason? low light camera the moon is bright! direct light from the sun with no atmosphere to block it out and with the low light would erase stars.
Another large piece people often use is, well if we made it on July 20, 1969 and we have better tech now , why is it so hard to land there now. Safty is the main reason, cccp had beat us into space, on every thing prior this and at the times it was deemed important for the US to make the next big leap "the moon". As anyone that was part of that will tell you, there was only a small chance of making it, and it was almost scrubbed on many accounts, including landing with less then 5 minutes of fuel.
After manned space flights became more normal, NASA had a string of bad rep plus disasters that all but stopped the program.
I beleive we landed, and i beleive those in the craft bought a lotto ticket and got lucky. Had safty standards been as high then as now, we would not of went to the moon.
I also feel its not really a great feat of the US, but a great feat that mankind in general was able to reach out from our tiny little rock and touch what no one other of our species had.If you reasearch the conspiracy theories you can normally find a non government or nasa person that can completely shoot that theory full of holes.
And Mars will be much diffrent cause its many countries working togather not just one, so the conspiracies will be lessened. (but still there)
Hope it helped.