Learn Russian: Я меняю свои привычки для того, чтобы жить по-новому. - I am changing my habits in order to live in a new way.

It is an adverb of manner meaning "in a new way," derived from the adjective новый (new). To form these adverbs, you add the prefix по- and use the dative singular neuter ending of the adjective. Since новый is a hard-stem adjective, its dative neuter ending is -ому, creating по-новому. Always remember the hyphen!

Why use для того, чтобы instead of just чтобы?

Both mean "in order to," and you could absolutely just say чтобы жить по-новому here. Adding для того, before чтобы makes the sentence sound slightly more formal and puts a stronger emphasis on the purpose or goal of the action.

Could I use мои instead of свои here?

Yes, but свои is much more natural. In Russian, when the possessive pronoun refers back to the subject of the sentence (here, я), it is heavily preferred to use the reflexive pronoun свой (in this case, plural accusative свои). Using мои wouldn't be grammatically wrong, but it sounds clunky to a native speaker.

Why is меняю (imperfective) used here instead of a perfective verb?

Меняю is in the present tense, implying an ongoing process right now ("I am changing"). If you used a perfective verb like поменяю or изменю, it would shift the meaning to the future tense ("I will change my habits"), because perfective verbs cannot describe ongoing actions in the present.