I establish first the common ground,not even touching on hot button topics. I find anything we can agree upon "those are cool shoes" and start up a normal conversation void of any intent. If people have their guard up "convincing" is never gonna happen. Too many want to try and force their opinion on others through weight of logic and debate. I find that very lacking and unfruitful. I get to know them,make them laugh,listen to them,ask questions,actually care to get to know them. The little personality traits they display will help establish a raport and know your angle of approach. They will see I am funny and want to make them laugh and have things in common. No rush to convince them of anything. Its like you dont rush trying to get someone to sleep with you by pressuring them. You make the experience pleasant and make them want to engage while you pick up little things about them that will help you get your point across. Its easier to convince a friend to do something than an enemy.
Rush nothing and let things play out, if the correct opportunity doesnt arrive then let it pass. When or if you see them again they will remember you in a positive light and you can continue in that way. When the time is right start with the smallest of things to get them to agree with you. Something or anything you know they will like "those are cool shoes arent they?". Them agreeing with you on the smallest of things primes the pump for them to agree on larger. Their mind is already agreeing with you and sees you as friendly. Then ever so slowly start to build on that, all while being content to walk away having "proved" nothing yet always building towards it.
As some would say, be wise as a serpent.