Don't expected implementing Git will resolve all your problems. If your team members use Git as well, you will still have problems. Following, 4 Git workflows.
Use only Master branch, maintain it clean as possible.
This workflow dont explores Git features, but is one option.
Every featured is turned in a branch. When job is done, merge it with master.
My favorite. Create branchs for releases, features and bug fixies. Take a look:
There are other details here, i recommend you study this flow. If you use SourceTree, there is a button "Git Flow"
This flow require a team member make a pull request when your job is done and the repositorie owner decide between accept or not. Like GitHub works.
Leave a comment if you use other flow.