I would avert the JFK assassination and have his administration be conciliatory and diplomatic with Khrushchev and Co. I would also have the JFK/LBJ administration hammer out a peace accord that includes a South Vietnam, probably with a leader who's pro-West and not a corrupt shithead like Diem. This averts a lot of the mistrust of the government that permeated the 1970s and 1980s.
JFK dies during his second term of Addison's disease, and LBJ doesn't run in '68 or runs and loses the primary, as the RFK assassination is also averted (let's say Oswald's a lousy shot or a bee stings him as he's shooting and he doesn't get a second chance, and Sirhan Sirhan gets arrested before he gets his shot at RFK.) RFK gets elected in '68 and Nixon goes away.
During this time, RFK gets landmark civil rights legislation pushed through, generating unprecedented equality before the federal law, and he also hammers through an improved public housing and Europe-style health care. He lasts two terms, and America, sick of the Kennedy dynasty, elects Ronald Reagan in 1976 on the back of decreased government regulation, more freedom and greater global influence, only phrased much more nicely.
It appears the Reagan revolution takes off four years early until stagflation hits and Iran goes to hell. The people blame Reagan and vote him out on the back of a counter-revolution and vote in a liberal Democrat - probably not Ted Kennedy, but Mondale is a definite possibility. He serves two terms, and with significant education reform and bolstered health care, homelessness and incarceration drop to below world averages and private prisons are unheard of if not outright banned.
Furthermore, in the 80s, restrictions on campaign contributions are passed, with "soft money" banned or severely restricted and no Citizens United-style decision. This opens up the chance for more independent candidates to be elected to Congress and to state governorships.
AIDS is managed right off the bat as sex education is entrenched in all 50 states, and with no "Evil Empire" speech, the Democratic Party is happy to sit back and passive-aggressively let the USSR implode. The military is reduced as European countries are expected to shoulder some more of the burden of defense, with some countries instituting Switzerland-style service. America has presence all over the world but its role in actual altercations is reduced to peacekeeping and keeping people like Saddam at bay.
George W. Bush buys the Texas Rangers and they end up being a losing ball club that moves to Vancouver, BC after investors buy them.
A few things that change - the 26th Amendment never passes and all but a handful of states lower their voting age to 18 by referendum or legislative action. Some even lower it to 16, such as Vermont or Hawaii. The War on Drugs never happens and the switch to a rehabilitative approach is much swifter. Barack Obama ends up staying in the Senate and becoming very popular; he is disinclined to run for President but is appointed a federal judge. The Republicans become something of a permanent opposition party, only coming into power every couple of decades and being booted out when they threaten Social Security or public health care.
Organized labor is stronger and has an entire section of the Democratic Party. Also, without the AIDS stigma, homosexuality is more tolerated and states pass laws allowing civil unions in the 90s with full-on marriage being more of a technicality later on. With more of a safety net, people of different races don't end up segregated as much, and with police less militaristic, there's reduced brutality toward minorities - it still exists but is less pronounced. Full-on acceptance of transgender people happens sooner, as a heavyset bearded man graces the cover of Vanity Fair in 2010 with the caption "Call me Chaz." And he still can't dance worth a shit.