A word or phrase to describe people who are practical and "care more about themselves"?

Machiavellian describes someone who manipulates situations to their benefit, using human emotions when suited, and dropping them when not.

An opportunist is "someone who takes advantage of any opportunity to advance his own situation, placing expediency above principle"; ie someone who stoops to almost anything to gain an advantage.

Ruthless means, in part, "not thinking or worrying about any pain caused to others"; for example, She ruthlessly pursued her ambition, letting nothing get in her way.

A selfish person is "concerned with oneself or concerned with one's own interests, especially to the exclusion of others."

Per previous answer, a Machiavellian person tries to achieve goals by cunning, scheming, and unscrupulous methods. If you want to suggest lack of scruples, opportunist and Machiavellian do so. If you don't want to suggest that lack, ruthless and self-serving may be a tiny bit better. (I slightly prefer the term self-serving to self-centered in this context.)