One word for an activity a person does to get close to someone

Is there a single English word for an activity a person does to try and get closer to someone? This someone might be:

• Someone they get a crush on, so they're like trying to be her/his boyfriend/girlfriend.

  or:

• Someone they want to know deeper. Like what a teacher does to her student, a mum does to her kids, a friend does to another friend.

In Indonesian we would call it: pendekatan.

In English, I can only think of:

• trying to get closer (which is not one word)
• approaching (which is merely translated to English)
• flirting or seducing (which is not the kind of approaching I'm looking for)

I suppose there is a more "English" word for this. (?)

It might be a little old-fashioned but I think "courting" is probably the most appropriate choice. You might prefer (the also old-fashioned) "wooing".

The act of undergoing such activities is called befriending:

befriend — make friends with: to be friendly to somebody, especially to somebody who has no friends and needs help

Befriend applies in each of the various situations you described. The relevant adjective to describe someone acting this way is simply, friendly:

friendly — affectionate and trusting: characteristic of or suitable to a relationship between friends

And there are various idioms and sayings that describe the act or intention itself:

making friends — form a friendship, foster cordial feelings, as in I hope Brian will soon make friends at school , or She's done a good job of making friends with influential reporters. [c. 1600]

warm up to — to become more fervent and earnest toward someone, something, or a group

It is worth noting that someone trying to befriend another person does not imply the intent is reciprocated:

Ugh, John keeps trying to befriend me.

There is one more word you might want to consider:

ingratiate (v.t.): to establish (oneself) in the favor or good graces of others, esp. by deliberate effort: to ingratiate oneself with the boss.

Synonyms include: curry favor with, cultivate, win over, get in good with; toady to, grovel to, fawn over, kowtow to, play up to, pander to, flatter, court, wheedle.

In it's strictest sense, it is to seek or buildup favor in another, and can be used this way. However, it has come to have a negative connotation of dishonesty, or a way to climb up a social ladder.