People who've done you a favor tend to like you more and do more, so start with small, interesting, easy favors.
The idea
Ben Franklin won over a political rival by asking if he could borrow a rare book the man owned. The two became lifelong friends. That's now called the Ben Franklin effect: once someone has done you a favor, they're more inclined to do another. Favors build goodwill and make people feel invested in the relationship; once someone has put effort into you, they see you as worth the effort. Franklin's ask was perfect because it was interesting, easy to grant, and flattering, since it implied the man had a collection worth borrowing from.
How to do it
Before asking, check: is it interesting? Flattering? Easy to fulfill? How will it make them feel?
Sequence small to large. Ask for the time before you ask for directions.
Give them an easy way to say no without having to say no.
Let a small yes lead to the next conversation.
Example
I usually breeze past people with clipboards on city sidewalks. One day I got roped in. To my relief, he didn't ask for money, just my name and email for a petition. Sure. Then came the emails, and eventually, yes, the ask for a donation. By then I was a lot more inclined to give. One small request at a time.
Script
Instead of "Want to meet for coffee?": "I'll be in Dupont Circle on Friday and thought of you. Is your office still there?" If they're interested and free, they'll invite you.
Watch out for
Opening with a big ask. Nobody lends a stranger a million dollars because a book once worked for Franklin.
The IKEA Effect: People care more about what they helped build, including relationships, so invite contribution instead of doing it all yourself.
Asking for Help Is a Gift: Helping lights up the brain's reward centers, so never asking for help denies the people who care about you a genuine feel-good moment.
Kill the Vague Ask: Say what you want, how long it takes and why them, and build in an easy out so that "no" is safe.
Avoid the Coffee-Request Trap: Instead of 'want to get coffee?', mention you'll be near them and let them invite you, or offer what you actually want to give.