Some Carter's problems were his own faults and some were out of his control.
If he had came to Washington more willing to work with the members of Congress. Through more bones to the Democratic leaders in Congress early on and made some of their top priorities his top priorities during his first few months as president, I think they would have been more supportive of him when things started to turn south for him in late 1977/early 1978. But he truly believed he was going to reform things in D.C. and cut a lot of federal spending as a symbolic gesture that Washington was going to "tighten their belts" during those very bad economic times like many Americans were having to do in their own households. But the Democrats in Congress were not feeling that and probably would have preferred him to spend more by implementing New Deal style relief programs like the Works Progress Administration, the Civilian Conservation Corps, and the Farm Security Administration for example. President Carter was also a micromanager (he famously chose not to have a Chief of Staff early on in his presidency). If he had had someone connected to the "Washington establishment" in his inner circle in the West Wing, he might have navigated the political waters better early in his presidency. He should have brought in former DNC Chairman Robert Strauss to be his Chief of Staff (he was Carter's U.S. Trade Representative). He had both Carter's respect and the respect of the congressional Democrats.
But for the most part many of the things that happened during his presidency were out of his control. The bad economy and stagflation began under Nixon's final year and things just got worse through Ford, Carter and for Reagan's first two years. The energy crisis, the hostage crisis, those things would have happened regardless of who was president. Those problems were simmering in the late 60s and early 70s. They just boiled over in the late 70s. If Carter would have been more hawkish about the Iran Hostage Crisis, I can see a sort of patriotic feeling develop to create a "rally around the flag" mentality to get the American public to stand behind President Carter long enough to get him enough electoral votes to beat Reagan in 1980 (remember that Reagan won many of the states he won in 1980 with less than 50% of the vote in those states). With more support for the Commander in Chief maybe there is less interest in John Anderson's independent candidacy, which also helps Carter in 1980.
That gets us into a 2nd term and then Carter can focus more on his legacy. He nominates the first woman to the U.S. Supreme Court (likely Shirley Hufstedler). With Carter re-elected the Iranians know that he will be there for 4 more years. So negotiations to release the hostages can continue and are more favorable for Carter's State Department now with the Reagan team no longer secretly trying to sabotage them. Carter would have continued to give aide to the mujahideen in Afghanistan in their war with the USSR. After Brezhnev's death in late 1982, I can see Carter by then in his lame duck period try to explore more diplomacy with the Soviets. He would not have called the USSR an "evil empire" in early 1983 like Reagan did and he wouldn't have been so thirsty for a "Star Wars" like missile defensive system like Reagan. Maybe without that extreme anti-communist rhetoric, Yuri Andropov would have been more willing to listen (though I doubt it) before his health started to fail. But because of Andropov's and Chernenko's poor health, I don't expect Carter could have done much when it comes to real arms reductions. He at best could have laid the grown work for his successor and Gorbachev to ultimately do the arms reductions we'd see in the late 1980s. Carter would at least not have put us in a position that nuclear war was such an extreme threat as it appeared to be in 1983.
Carter would have kept Paul Volcker as the Fed. Chairman and the economy would have still began to improve by 1983 and 1984. Now could Vice-President Mondale have won in 1984. That's certainly debatable. The Conservative Coalition I think would not be as strong with a Reagan loss in 1980. But they would still be there and hungry for power. Maybe they do well enough in the 1982 midterms to win control of the senate. But I can see a bloody battle during the 1984 Republican Primary between Bob Dole and the ultra right-wing conservatives and George Bush and the center-right of the party. Dole probably wins because he would've been a more familiar face to the public as a sitting senator during Carter's 2nd term. So he would have been on the Sunday morning shows more, he could have introduced conservative legislation and initiatives as a sitting senator. He just would have been more visible opposition to the Carter White House during Carter's 2nd term than Bush would have been. In a Mondale vs. Dole election in 1984, I would believe Mondale would win that race with an improving economy and with Dole not having the charisma that Reagan did, he would turn off middle of the road voters who may have personally liked Reagan.
After Carter leaves office in 1985, he returns to Plains, Georgia and opens the Carter Center in the late 80s and does much of the same work he's done the last 30+ years.