I'm not sure what region you're interested in, but I can speak about the UK, particularly England. I'll also try to address the experiences of both gay men and lesbians, because there are some interesting differences in how these groups were treated differently in the 50s and 60s. And I'm actually going to start with WWII, if that's okay, because it makes the 1950s make more sense.
WWII made Britain in many ways a relatively okay place for gays and lesbians. The war created very homosocial environments among men in the armed forces and women on the home front. The need for women in the workplace, the lack of men due to military needs, and the removal of children from London all altered family structures and earlier gendered expectations. Additionally, the blackouts created an optimal environment for cruising and quick hookups. The darkness offered a measure of protection for those searching for same-sex partners. The nearness of death created an urgency that encouraged impulsive couplings. For a humorous but important take on wartime sexual freedom, see Quentin Crisp's film The Naked Civil Servant.
After the war, the men returned from the front, the children came back from the countryside, women were encouraged to become homemakers again, and the lights came back on at night. The postwar ethos of conservatism batted back at wartime morality. Urban and suburban housing increasingly focused on the (presumably heterosexual) nuclear family rather than single workers in the city. Parenting styles become an explicit concern at this time; it was in postwar America that Dr Spock's Baby and Child Care was published. It was also at this time that the overbearing mother was posited as a potential cause of male homosexuality.
In the 1950s there is also a proliferation of sociological scholarship with presents gay and lesbian people as a social problem. Queer people, especially men, were seen as on the periphery of the institution of the Upstanding British Family and Way of Life, and even as a threat to it. "Chain arrests" were relatively common--one gay man would be arrested for soliciting sex or another offence under the Labouchere Amendment, and investigators would offer the accused a lighter sentence if he identified the other gay men in his address book. In this way, whole circles of friends could be imprisoned after only one of them was arrested.
This was the heyday of polari, a slang system which allowed gay people to have potentially damning conversations without being understood by the wider public. You can hear an example of polari in this sketch featuring the characters Julian and Sandy in the comedy show Round the Horne. Note that, partly due to the show letting straight people in on the key to the slang, polari went into decline in the 1960s.
In short, gay men felt embattled. Lesbians were more able to fly under the radar, as the law was less clear about female homosexual activity.
But there was hope for these people. The Wolfenden Report of 1957 recommended the decriminalisation of homosexual acts, with some restrictions. The Wolfenden Report wasn't arguing for a rights-based legalisation of homosexual activity, but rather that the law had no business telling consenting adults whom they could or couldn't sleep with. The report did not immediately result in a change of law, but it offered hope to gay men in particular.
The big event in this era was the 1967 Sexual Offences Act, which adopted the earlier Wolfenden recommendations and decriminalised sex between TWO consenting parties OVER THE AGE of 21 whilst IN PRIVATE. (Note that the age of consent for heterosexual acts was 16, and straight people could legally have sex involving more than two people.)
In many ways, 1967 was huge for gay men: it legally legitimised their sexual behaviours. For women, it was relatively irrelevant because lesbian sex had never been illegal. You can watch really interesting BBC documentaries from 1967 about the debates on the passing of the act here for gay men and here for lesbians. I highly recommend it, and I think it nicely illustrates that despite the official proscriptions on homosexuality, queer people went right on living their lives.
You might also be interested in the 1968 film The Killing of Sister George, which depicts a dysfunctional lesbian relationship. Of particular interest is this scene which shows the Gateways Club, a real life lesbian bar in Chelsea. The people in the background in this scene are actual regulars from the club.
I know I focused a little more on law and media than on daily life, but I hope this answers your question okay.
Sources: The links above for primary sources, plus...
Cook, Matt, 'Queer Conflicts: Love, Sex and War, 1914-1967', in A Gay History of Britain, ed. by Matt Cook (Oxford: Greenwood, 2007), pp. 145-78
Houlbrook, Matt, Queer London: Perils and Pleasures of the Sexual Metropolis, (Chicago: University of Chicago Press, 2005)
Mort, Frank, 'Scandalous Events: Metropolitan Culture and Moral Change in Post-Second World War London', Representations, 93 (Winter 2006), 106-37
hi! Could you specify which region you're interested in? Whether that's a country, province/state, city - the more specific the better for potential respondents. Thanks!
Check out Stonewall by Martin Duberman for a look at gay life in New York. There was a diversity of experiences. Some sought psychiatric help. Some found romantic partners, and managed to live together as roommates. Others went cruising, which consists of walking in public in known cruising areas, giving potential partners long, staring glances with the ultimate goal of casual sex. Stonewall looks more specifically at the bar scene in New York. Police officers were conducting raids on gay, lesbian, and trans bars at the time. People who were caught during the raids had their names published in the paper, and risked losing their jobs.
Finally, after one raid at the Stonewall Inn, bar goers refused to leave, and instead occupied the square outside. Police officers were unable to subdue the crowd, considered embarrasing at the time because police couldn't win a fight against a bunch of homosexuals. Stonewall became a watershed moment in gay rights, and a parade commemorating the event was held a year later.