How am I to get trust from people don't care about me or my wellbeing?

The concepts are largely unrelated. You gain trust by upholding their expectations through trials where other opportunities arose (eg siding with them without promise of personal benefit). They may then prefer to preserve such an acquaintance, or exploit you. It is up to you to seek & discern where you'd like to invest in loyalty.
At any rate, if you wish someone to care about you (and I'm saying someone rather than people intentionally, because there is no fulfillment from that shallow concern when you're in need), you'll probably have to expose yourself to hope for the same, and accept that it will still be difficult to attain. Family's usually a decent place to start.