Sometimes information regarding git are more detailed and easier accessible when using git-bash than using TortoiseGit. Therefore, I would want to use git-bash for very specific issues (e.g. creating a branch/checkout at the same time and commit uncomitted changes).
git checkout -b my_new_branch
git commit
Could I work with TortoiseGit without any problems afterwards? And vice versa?
Yes. tortoise-git is just a user interface to the command line tools which just interpret the .git subdir in your project's directory. You can use them both on the same project, though running actions concurrently might cause problems, I'm not sure.