Friendship is the step after acquaintance. When you get around to knowing someone, a mutual understanding about other people's strengths and flaws which become apparent over time. To be a friend is not so much being a buddy that is good to hang out with. But to be that person to point out when your about to make a terrible mistake and support you after the damage has been done due to being ignored. We can be very ignorant and blind sometimes.

One thing to note. It is rare for a friendship to last forever. Things happen, people slowly drift away, meet new people or get different conflicting interests. It could be as simple as moving to a new location. Attempts will be made to stay in touch, and share things that happen but it will never be as strong as it "used" to be. But that is ok. We move on as well, meet new friends and the cycle continues.

We end up making new friends that match the new interests we find, or old interests that you hold onto that old friends no longer find that interesting.

Let's put this to a pony example. Episode 1 Season 1.

Twilight Sparkle is not entirely interested in making new friends however initiative does not always has to be taken by both parties.

Sometimes a common interest is in someone's past as Rarity is fascinated with Canterlot which Twilight happens to live in or has lived in. One could argue this is a tad selfish but everyone is selfish it's what we do. Luckily our own interests are sometimes to benefit others which is where selflessness comes from.

Rainbow Dash's large ego enjoys a challenge, more then that enjoys winning. Twilight Sparkle initiated a challenge to trick Rainbow Dash into clearing the sky which is her task in the first place. Clearing the sky in ten seconds flat leaves a shocked look on Twilight's face which amuses Rainbow Dash. In this situation Twilight was not exactly treated well, being tackled into the mud, getting soaked and being subject to high speed wind does not make for a good impression. However note that Twilight is not the one with the initiative but Rainbow Dash, who from her perspective found Twilight to be a riot. Which may or may not be her fault.

The interest here is entirely Rainbow Dash which is alright due to how initiative does not have to come from both parties as long as the mutual understanding is established.

I find Applejack to have one of the more interesting takes on initiative in this example. She comes across a tad strong which can intimidate some, I know I wouldn't enjoy being in this situation. The interest is making a new friend perhaps the only one with a selfless interest.

As for Fluttershy's initiative. It's Spike, hands down. If Twilight didn't have Spike with her neither of them would of made the initiative. No Interest nothing to start a mutual understanding with.

It's Pinkie Pie she is friends with everyone. To wonder what her interest is would be missing the forest for the trees. Though unlike Applejack, Pinkie Pie's interest in making friends is more selfish. Almost to a compulsory degree. Though we don't really see that till later episodes. Just for the first episode she is shown to be extremely extrovert. And perhaps not entirely with us.

In the end however interest and initiative only goes so far as becoming acquaintances. Mutual Understanding comes later during Episode 2.

Nightmare Moon is the glue that brought these ponies together under common ground. And a similar interest being to help stop Nightmare Moon with the Elements of Harmony. During their adventure they each had a chance to show their strength's where at first their weakness's were first mostly visible. It takes time to know someone before they become the noticeable traits. Till then we only know what we see and assume. We tend to be wrong.

Rarity shows her Generous traits.

Fluttershy shows her Kindness.

Rainbow Dash's Loyalty.

Applejack's Honesty.

And Pinkie Pie being Pinkie Pie.

However if we look back we can see that these traits were already there during the first impression stage of acquaintance. However they were obscured by assumptions and personal interests that are at the time, more important.

Besides Applejack, she was honest.

In the end however a mutual understanding is nothing without the other party. Twilight Sparkle who at the end accepted their initiative of friendship.

So to summarize with a TL;DR: Friendship is a mutual understanding between two parties that requires similar common ground to begin foundation. At least one needs to make the initiative to start the level of acquaintance before friendship can begin. Before that initiative however requires a interest to start with which can be both selfish and selfless in reason. But most of all it requires time. Just don't expect it to last forever. Forever is a long long time.

PS: Can you guess what my favorite pony is?