Is this sentence right?

Well, they’re both right but they have slightly different usages.

“Important for me” is usually followed by “to do” or “to” some verb and the word “it” is an introductory word.

“To me” is okay on its own. As in “it is important for me to lose weight” and “I want you to come to my party; it’s really important to me.” In the last sentence, “it” refers to something else, the party. In the first sentence “it” is just starting out the sentence; it isn’t referring to anything specific.

I don’t know about any grammatical law regarding this, though.

“It is important to me” is proper English, but the statement is ambiguous unless you have already declared what “it” is.

“It is important to me.” is perfectly correct. “It is important for me” could be correct, but that it was important that you do something, not just that some issue meant a lot to you, and you would need to finish the sentence out. For example, “It is important for me to go running today.” or “It is important for me to get there on time.”

Let’s see the entire sentence. Then we can tell.

It is important to/for me that the people around me are happy. I think its to? is it?

Using @DominicX‘s example, “It is important to me to lose weight” vs “It is important for me to lose weight”, the two sentences have different meanings. “For” has an implied sense of urgency, whereas “to” is more abstract.

It is important to me to lose weight because I feel fat. (correct)
It is important for me to lose weight because I feel fat.

It is important to me to lose weight so the dress fits correctly.
It is important for me to lose weight so the dress fits correctly. (correct)

Correct: It is important to me that the people around me are happy.

Yes, “to” is correct in that sentence. In that construction, “it” is an expletive standing for the entire clause “that the people around me are happy,” which is the true subject of the sentence: ”“That the people around me are happy is important to me.”

The phrase “to me” simply qualifies “important”: to whom is it important? To you.

The “for” construction is quite different. Using the same example as others above, you would recast the “weight” sentence thus:

For me to lose weight is important.

Here the phrase “for me” goes with “to lose weight”—who is losing the weight—and not with “is important.”

Strictly speaking, the sentence also requires the present subjunctive: “be” rather than “are.”
It is important to me that the people around me be happy.
However and alas, the useful and elegant present subjunctive, though not quite dead in English, is certainly calling for its last rites.

In that sentence, definitely “to”