Most adults make friends at work, parties & gatherings, church activities, or if they have children they may bond with other parents over school programs, the gym or anyplace of where the same people are in regular attendance.
Meetup.com hobby/interests groups where people meet once or twice a month, evening classes of any kind, and even while on vacation.
Being in a shared environment/situation is oftentimes the launching point for having a conversation. I know on my cruise lines most people are seated at an assigned dinner table with other people they may not know. By the end of the week usually there is a little bonding between everyone.
Naturally how one defines "friends" is important. Most people aren't going to qualify as being "best friends" or almost like "siblings".
However there are plenty of folks where if one wants to maintain contact with periodically it can easily be accomplished (phone, email, and social media). From time to time extend invitations to events and gatherings.
Another thing these days is people are suspicious of those who seem to latch on to them too quickly. They'll tell their real friends; "I barely met this woman now she's calling me a couple of times a week, inviting me to go shopping, to her house for dinner...etc I wish I had never given her my number." You don't want to come off as being desperate for friends.
The worst mistake anyone can make is "rushing a friendship along". The best friendships occur over time. Sometimes you learn something about someone and you decide to pull back.
It's all part of the process.
Just about everyone has experienced having the "annoying friend" or person who wants to forge a friendship with them and they're looking for ways to avoid dealing with them. It helps to be observant to recognize when the other person has no interest in knowing you beyond a casual acquaintance.
A real friendship is (reciprocated) with calls, emails, and invitations. No one should always be "the one" initiating the contact. If it is try not reaching out and see how long it is before they contact you.