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
In this link, the last line which says "As an alternative, you can use a third-party build, test and release management application to get the latest version of test projects from the source control and run them automatically"
So now the question come is , can Jenkins take care of the whole process like to get the latest version of test projects from the source control and run them automatically ?