How can you get the trust in your new job as manager or responsible?

Don't be bossy around you're team. Be kind, be open minded, treat you're team equally. 

Be open, honest and fair.

When you say you will do something by a certain time, do it.

Neither avoid nor seek conflict.

Find out what really  happened to your predecessor, and respond to the team accordingly.  (If he or she was good and popular, and somebody in the team was expected to be promoted, then you have a high hill to climb.)

Bosses don't like surprises.  Neither do team members.

Be consistent.  Be firm.  Be kind.  Be reliable.