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

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.

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".

Perhaps cultivate

to give special attention to: to cultivate a friendship, to cultivate a hobby

It might be used in a phrase, such as cultivate a relationship with ...

Fraternize means to associate in with others in a congenial way. (Literally means to make like a brother.)

Consort means to associate or keep company with.

Both of these words often are used when describing an improper relationship (e.g., fraternize with the enemy), but they do not have to mean that.

For words that are more associated with flirting and sexual attraction, consider

Chase, meaning to follow or devote one's attention to with the hope of attracting, winning, gaining, etc.: He chased her for three years before she consented to marry him.

Pursue, defined as to follow persistently or seek to become acquainted with.