Ok, so, a lot of cis people who are trying to work on their cis privilege ask anyone that they see as trans, genderqueer, or gender non-conforming their pronouns upon first meeting them or being introduced to them. That way, they figure, they don't mispronoun anyone, and everything is great, and they get their good ally cookie. (A variation on this is only asking people whose name and/or appearance makes things unclear, which is more othering to the people who experience it).
Wrong. One, if you're only asking people who are visible to you, you may be a) missing the genderqueer person whom you read as having a binary gender and b) you may be outting a trans person in a situation where it may well not be safe to do so. (i.e. "She didn't ask Bill, or Susan, or Mary their pronouns, but she did ask Melissa..."). So, then, how do you go about getting people's pronouns so you don't mispronoun anyone, without othering anyone?
In informal situations, you can just wait to use a pronoun for anyone until either they give you one, or until someone you are absolutely certain knows and is using the proper pronouns uses one. Obviously, the later is still a judgement call, and if you're waiting on the former, you had better be absolutely certain that you don't just start using definite pronouns for people you assume are cis until they make a pretty definite statement (such as, "I'm a cis woman and in standing in solidarity with trans people..." or "as a trans woman, I experienced sexism like..."). Or, you can just deal with causing cis people a bit of momentary confusion, and ask absolutely everyone their pronouns.
In formal situations, it's a little easier. If you're going around a circle, for instance, have everyone when they give their name, give their pronouns. Once again, it's a bit of momentary confusion the first time for a cis person, but it prevents trans/genderqueer/GNC people from feeling like their identities aren't legitimate and their Preferred Gender Pronouns aren't some sort of weird thing.
As a tangent, cis people, please do not say you don't care when you are asked your pronouns. You're not showing yourself to be cool with trans people, you're showing that you have no clue what it's like to be mispronouned all the time. I'm willing to bet that after they say that, everyone still goes and uses the pronouns fitting the sex they were assigned at birth. There are actually genderqueer people who prefer alternating pronouns, or genderqueer folk who otherwise want their pronouns varied, but when a cisgendered cissexual says that they don't care about their pronouns, they're showing off the fact that a) they don't get mispronouned so b) they aren't going to be bothered if someone uses a different pronoun for them, because they don't have to deal with it all the time.