You need no or a much shorter Vietnam war. And likely no Oil Crisis of '70es. (although this will hurt Soveit ecconomy much, but maybe things go different for them with Krushchev remaining instead of Brezhnev) And more space competition from Soviets. Khrushchev stays in power for several more years. Korolev survives his surgery and either makes a N1 that can work or admits is hopeless and allows Glushko to start building Vulkan (Energia predecessor that never existed outside paper IRL) by the end of '60es.
For American political scene, its tough to tell. Even if Kennedy isnt assasinated, we dont know whether he would have been reelected. And for his brother to win presidency afterwards... JFKs presidency would have to be flawless.
So a rough sketch of the timeline. Appollo 11 as IRL. Appolo program not shortended, Appolo Aplications program not cut. With Saturn V is continued use it is improved to be a more modern more efficient heavy lift launcher. With it the shuttle can be a small and affordable space plane just for crew launches. That makes it far cheaper than in OTL.
Soviets get a heavy lift vehicle by 1970. and have a succesfull Moon landing by 1971-2. They go both for Salut (to parry USA Skylab) and decide to go for a moonbase, to make it up for the fact that they were second on the Moon.
Basicaly, both space programs go as smoothly as possible following optimistic IRL plans. By 1980. both USA and USSR have Skylab and Salut succesors on orbit, and are plannig a modular MIR type stations. Both are planing manned Lunar outposts, with one having actualy launched the hardware. Spaceplanes for crew transportation are finishing development. And expandable launchers are being continously refined and pushed for price efficiency.
Buildup of hardware and infrastructure troughout '80es. In '90es first GEO manned outposts. (Although there is no need or reason for large number of GEO manned outposts Clarke always predicted. Its irronical, since he was a radar engineer in WWII he had great distrust of electronics, and couldnt belive complex enough systems for big GEO relay stations could function unmanned. With weather and telco GEO satelites from IRL, there is very little need for GEO manned space stations)
By end of '90es first Mars mission is launched, succeds and returns. And TMA1 is discovered on the Moon
Since, book and "Lost worlds of 2001" clearly stated. They werent ready for Jupiter/Saturn. They were only forced to figure out how to accomplish something that difficult becasue a freaking alien artifact on the Moon transmited something in that direction.
Edit. RE: Petike's post. While his version is stretching it a bit, my is a bit more balanced. AND AND AND. Without Vietnam and Oil Crisis and recession of the times. And with both space programs running with far less bureaucratic waste and inefficiency, with silly things as Shuttle cost lies being seen on time. It was possible, barely, but possible. There was no scientific problem to reaching Mars in 1980'es or '90es. It would have cost a lot, and with current inefficiencies in space program it would have been directly unaffordable. But with a well managed program all engineering difficulties and issues could have been solved. NASA personnel saying "we don't have scientific capacity to send people to Mars" is 101% just sower grapes, consoling themselves that NASA didn't do anything for last 30 years. Radiotion shielding is possible, just heavy, lander suitable for Mars is possible... it all comes down to mass in LEO needed for a sucessfull Mars run. Just $$$ or waiting to build a LV that doesnt eat money for fuel each launch.