Github - Local repository already exists. How do I create a remote github repository and sync to it?

Summary of what I did

I frequently encounter the following problem:

I create a folder for a project on my local machine and start working
Later, if I want to keep this code longer term, then I would like to be able to add it to version control using Git

Note: Rust and Cargo: This problem occurs by default when using Rust and Cargo. Cargo (the Rust package/project manager), by default, will create a git repository when creating a new project with cargo new. This means that, if you have already created a git repository, for example via github.com (or others) there is no obvious way to synchronize the two.

One solution is to:

create a Github repo (on Github, Gitlab, Bitbucket etc)
clone this repo from the cloud to a new directory on my local machine
copy the files from the existing directory to the new one created by the git clone step above
commit new code in the new repo and push the new changes
finally, delete the old directory

This seems like quite a common solution, but as far as I know there is no convenient solution.

Questions

How do I push my work on branch master from the local repo to the remote?
How do I pull down the new files from the remote repo to the local?
Essentially I want to sync the 2 repos without going through the whole process of making a new folder and deleting the old one as I described above. How can I do this?

Solution

This is a much easier solution:

cargo new my-project-name # FYI: cargo creates a directory with a git repository
cd my-project-name
git remote add origin https://github.com/github-username/my-project-name.git
git add .
git commit -m "initial commit"
git pull --rebase origin master
# ... now resolve any conflicts ...
git push --set-upstream origin master

This is not specific to Rust + Cargo, I just use this as an example.