気に{入|い}る
is a super common expression meaning that you like something or that something is your preferred one. It implies you use it a lot.
このペンすごく書きやすいんですよ、だからすごく気に入っています。 This pen writes super comfortably, so I particularly like it.
このバソコンは私のお気に入りです This PC is my favourite (I particularly like it, and use it a lot.)
It can only be used for things that the speaker can use
あなたのシャツが~~気に入っています~~ あなたのシャツが好きです
このバーすごく気に入ってるんだ。 I really like this bar. (I go here often)
Because it implies you use it a lot, you cannot use it when it is your first time. e.g. going to a cinema to watch a film: この映画が~~気に入っています~~
With 〜た, implies that it has really gotten to you and you want to use it a lot from now on.
この腕時計すごくいい、気に入りました。 This watch is super nice. I like it, and I'm planning on using it a lot.
The thing you plan on using it THIS SPECIFIC ONE, so cannot use it on consumable items このクッキ~~気に入りました~~！ ✕ このようなクッキ気に入りました！ ○
When used in negative, can imply there is some aspect of something you just really can't deal with. 田中さんいつも若い女の子とばかりいるよね。なんか俺田中さんが気に入らない。 Tanaka always hangs out with young girls. He really gets on my nerves.
気に食わない
same way as 気に入らない, but it is morerough, and hencemale
なんか俺田中さんが気に入らない。 ○ なんか俺田中さんが気にくわない。 ○ Takaka really gets on my nerves.
気になる
To be curious: Something caught you attention, and you you want to know an answer.
この箱の中身が気になります。 I really wanna know what's inside this box
It can only be used for oneself! 田中さんは箱が~~気になる~~
Used when you might be romantically interested in someone
ジョンのことめっちゃ気になっている。 I>Ik zie Jonathan wel zitten :3
in negative, shows something didn't really bother you
袖の穴が全然気にならない。 I'm really not that bothered by the holes in my sleeves.
気にする
Denotes worrying about something. You can't stop thinking about something because you care about it. Is used for other people, unlike 気になる target with を.
田中さんがちょっと太ってきたので、体重気にしてるんだよね。 Tanaka has become a little chubby, so he is thinking a lot about his weight.
You don't use it to talk about yourself. not wrong, just weird.
私は体重を~~気にしています~~。 △
Has the nuance of worrying: Therefore, not used for positive things. In the following sentence, it implies something went wrong at the game, and that makes him worried.
あいつは昨日の試合を気にしている。 He cares about the game yesterday.
日本人ってさあ、対面ばかり気にしてるよね。 Japanese people worry too much about what other people think of them.
政府は国民の何人死のうが全然気にしていないよ。 The government really doesn't care how many people die.
気にしなくていいよ Don't worry about it!