How easily do you give up on people?

When it comes to giving up on people and cutting them out of your life, you're probably a bigger loser if you repeatedly allow them to violate your trust instead of showing them the door. That being said, I don't think I would call you a loser, necessarily; you should decide for yourself what your standards for dealing with others should be, and only alter them if you find it necessary to do so. Of course, if you have standards, you're going to suffer a number of letdowns in your life when it comes to dealing with people; some people will exceed your expectations, others will fall short, but most, if not all, will be flawed individuals (like you, in fact) and you should always expect someone to disappoint you at least once in your life.

I never give up on people. Just follow your dreams!

"Am I a loser or childish if I give up on people easily if they lie or break my trust"

This truly depends on how long you've known the people in question. For example if it was say, half of your life and they broke your trust or lied you may not like it but you wouldn't give it much thought so long as a habit wasn't made out of it. That's how life works.

It depends on who they are. For example, If this person was a sibling who stole from me, I'd eventually forgive them, though I wouldn't trust them to be around my property. If the person is a friend or significant other who stole from me, I'd shut them out of my life permanently.