Normal workflow is either to clone an existing repo, do work, then pull + push to remote. Missing in diagram below is adding the remote to the new local repo.)
flowchart LR;
A[New directory]-->B[git init .];
R1[(.git created)]-->B;
B-->C[edit];
C-->D[git add .];
D-->E[git commit -m msg];
E-->C;
E-->F[git remote add <remote> URL];
F-->G[git push remote branch];
G-->R2[(remote git repo)];
Git can be used to sync with two different remotes, using pull/fetch and push. Branches can also be used to track work done on each end.
flowchart LR;
A[Remote repo A]<-->B[Local repo in .git];
B<-->C[Remote repo B];