This is a good question.
Everyone is a bit different. SOme people, like my parents. I don't bother. They were independents once upon a time (when I was young), now they are old and they are Fox news bots. They just repeat whatever is said on fox news. It's completely useless to speak with them. The older they get the more Foxnews robots they become.
With the rest of people, I try to find the issue the they can RELATE TO.
Are they big into privacy? *BOOM* NSA, snooping, etc. My cousin freaked out at the amusement park in tampa where they wanted you to look into the camera and do a facial recognition while submitting his ticket. He refused. (This was a couple years ago). Hit him on privacy and stuff.
Do they think police are bullies? Unjust? Assholes? (well...give him a few AF articles, and make sure they are CLOSE TO HOME, in his neighborhood or nearby).
Do they hate stupid Pot laws?
Do they hate war?
Do they think your milk or food is full of B.S chemicals? Mansanto haters? They like ORGANIC foods???
Do they think we are over medicating kids and turning them into zombies? Well well, there's plenty of Big Pharma evils to discuss (school shootings and more).
Do they hate taxes? Or Government waste? Well, there's unlimited things you could discuss.
I try to FIRST find the CONNECTION with people, so they AGREE with you. Now they are not DEFENSIVE, they are "OK, this guy, he has credibility, I trust his opinion. I agree with him."
Then I try to dig a little bit further so if they are open to seeing the light....Soft selling a little bit...Just give them a little. I won't go from, "Man taxes are too high right? Hey that 9/11 shit was an inside job and the government puts chemicals in the water that will make your dick fall off, and they will stuff you in a FEMA camp if you complain about it....I'm serious...check out this video..."
That's not going to work. And I've seen people do that. I was at a Ron Paul meet up many years back, and this guy was a Phd or Md of some kind and he talked a lot of sense....People were paying attention to him. Then he went on a HUGE RANT about CHEMTRAILS and got really intense about the Planes and the chemicals falling from the sky..and he wouldn't shutup about it, and then people walked away...like....WTF is that guy smoking...In his first meeting with tons of people he built up credibility and then lost it all and was tuned out.
I have a friend who read the hunger games and felt uncomfortable that maybe that could be a terrible reality one day. She's a liberal, anti-war, gay rights, into organic foods etc. So I read it as well and it was a good to build rapport. She was open to hearing more, and every day I have her sounding more and more libertarian. She seeing beyond the left/right paradigm. I know I cannot agree with her on every point, She hates guns and she cried for days on Sandy Hook. I don't need her to agree on every point, as long as I can wake her up to all the other stuff going on it's still a win. I don't need a libertarian clone, I just need someone who can see beyond the MSM BS that we are fed everyday.
Some people who follow their politics closely and listen to Hannity and Rush day and night...They may not be open. I think they enjoy the Red team Vs Blue team B.S. just too much. It's like a football game, so the truth doesn't matter as much as winning.
However, I have met people who listened to Rush everyday for years and years, and one day found themselves a bit lost by President Bush's actions, and Ron Paul walked in and something clicked. They began to see things in a totally different light, and of course I was adding my 2 cents all along the way.
The great thing about loving liberty is you can usually find SOMETHING that a Liberal or Conservative or Independent will agree with. Use that as your way to find a commonality, and then gauge if they are open to finding out more....Open to seeing the truth and waking up.