It seems like most of the stuff he is credited with was actually accomplished by LBJ. I know he had the Cuban Missile Crisis, but I have heard mixed reviews on his handling of that as well.
What exactly did he accomplish as President?
On the bottom of a list you could say that he avoided Nuclear Armageddon. His better accomplishments however include the creation of the Peace corps, which still exist today. He also had the accomplishment of putting the first man to the moon, igniting the Russia v. America space race and forever changing humanity's view of space. Further along humanitarian accomplishments, Kennedy created the USAID, which coordinated aid to foreign countries, mainly in Latin America. You can also say that JFK got the world on it's feet during his speeches.
I'm going to answer this question like an historian of the Cold War, because most Cold War historians put Kennedy (at best) in the middle of the pack for the U.S.'s greatest presidents. I would rank him far lower. I'll get some of the sources out of the way right now - McNamara's memoirs, Acheson's memoirs, Robert Dallek's biography of Kennedy, Naftali and Fursenko's "One Hell of a Gamble", Paterson's edited compilation, and almost the entire corpus of Stephen Rabe.
First, I'm going to deal with some of the things that have been said. Man on the Moon - No. Kennedy died two presidents before any man on the moon. He didn't ignite the space race, the Russians did in 1957, with Sputnik. Like so many other things, Kennedy simply took an Eisenhower plan, and tinkered with it. Civil Rights - This is almost all LBJ. Kennedy actually waffled a great deal on Civil Rights, and watered down his 1963/64 Civil Rights Act quite a bit. MLK and many in the Civil Rights movement felt (correctly) he'd deceived them.
So, Cuban Missile Crisis. Very quickly, that crisis happened because John and Bobby were obsessed with overthrowing and killing Fidel Castro. I invite you to peruse any FRUS volume that deals with Latin America during JFK's presidency and examine just how often Castro and Cuba were brought up, no matter what the topic was at the time. Now, did John and Bobby eventually clean up their mess? Yes they did, with Bobby getting special credit. Still, if I keep a fire I started from burning down the neighbourhood, I'm not really a hero so much as a jerk who got lucky.
As for Kennedy's "Alliance for Progress" with Latin American nations, here we have a very fun example of Kennedy historiography. The AFP failed, not because Kennedy was ill-intentioned or Teodoro didn't do his best administering aid, but because the economic and political agendas were misaligned. Pro-Kennedy historians and journalists tend to blame LBJ for the Alliance's failure, just as they blame him for Vietnam. But here too, Kennedy was effectively adopting an Eisenhower program (the Social Progress Trust Fund) and tinkering with it (and as regards Vietnam, Kennedy clearly increased the number of "advisers" but again kept fairly close to Ike's policies). The point I'm making is that Kennedy's Peace Corps was his greatest standalone achievement. Otherwise, in his time he accomplished very little concrete gain. Unless of course the Clean Air Act and a do nothing commission on the status of women are huge wins?
As Woodward argued, Kennedy had enough time as President to make mistakes and learn from them, but not enough time to actually do anything about it. I'm in no way arguing Kennedy couldn't have done some amazing things, he well could have. But with the Bay of Pigs, compromising on Laos, pushing the Cuban situation to the point of a near nuclear war, having no end of headaches with Adenauer and De Gaulle, refusing to do what was necessary with the Bretton Woods system, bungled Middle East balancing, and really not knowing how to deal with African leaders (like Nkrumah), Kennedy was simply not prepared or in his time able to deal with everything.
As a concluding aside, my area of specialisation is Eisenhower's foreign policy, and I have to say outright that a great many of the problems Kennedy encountered were in large part due to Eisenhower and Dulles (until he died) treating international issues like spinning plates - trying to keep them spinning rather than gently deal with them.
As for accomplishments? Peace Corps, resolving CMC, genuinely attempting to deal with problems Ike and Dulles had put off, settling on Laos, and (rightly or not) inspiring a great many people for a time. Kennedy was beloved, not in the wink and chuckle way Clinton is/was or in the solemn admiration manner accorded to Lincoln, (or even the grandfatherly respect for FDR), but in a very heartfelt personal manner. Think for a moment about America's presidents, very few accomplished simply being beloved, and none are beloved in the way Kennedy has been. That is an intangible, but very real accomplishment.
Jump start civil rights, began a war on poverty, essentially pushed for the rights of all Americans, He was the president for the people, who addressed the concerns of the new free-thinking generation.