Stash-Git is an Atlassian git management server while SourceTree and TortiseGit are both just clients with a GUI.
They all operate behind scenes the same way, using git.
You can either use Test Complete's built in SCC integration, or use either of the clients mentioned above. I personally prefer to not use the integration as we also keep a lot of non TestComplete assets in the repo and am more of a CLI guy myself.
As far as how to commit your code that is a very broad question. Has your IT department setup or provided a repository for you to use? If not that would definitely be the first step.
I would also make sure to research how git works (branching is your friend). Atlassian has a great set of tutorials which will also provide information on how their management projects work: https://www.atlassian.com/git/tutorials
Standard Git documentation: https://git-scm.com/docs/gittutorial
Finally things that you do and do not want to have tracked by your SCM provider unless you like merge conflicts:
http://support.smartbear.com/testcomplete/docs/working-with/integration/scc/files.html
Besides that if you have a specific issue or question about setup and usage I'd be more than happy to try an assist.