How do you get your new team to trust your leadership when they were very attached to their former leader?

Tough circumstance. It is hard to come in and replace a former manager particularly if he has good relation with staff. The issues to be considered is why are you replacing them, is there a problem with team performance or is it natural progression.

If it is a performance issue and your job is to address lack of performance and demand output from them you are not going to be liked if the team is lazy, however ,this is contrary to human nature people generally want to be productive and contribute. Your job is to create the opportunity for improving performance and rewarding accordingly.

If it is natural progression use the 3 R's

1. reason with them, if that doesn't work,
2. reciprocate as them to put themselves in your shoes to appreciate your interests and if that doesn't work
3. resources take resources from them

good luck.

This is something very important for a Manager/team leader. 

My suggestion are as below.

1. It is always recommended to have more team meetings and discussions specially when you are new to the team.

2. Encourage the staff and appreciate them on there productive ideas or suggestion.

3. Give them freedom to freely express the views and concerns and if needed take corrective actions to solve them.

4. Having team outings will be an added advantage.

5. Never criticize the work of previous Manager/leader, instead if it was done wrongly then you should take a different approach and prove your point by getting it right.