unable to checkout branches and proper procedure for creating a new branch based off of another one

Say there's a Git repository with two branches: default (which is the default branch) and branch. I want to checkout branch and start working on that but am unsure of how to do it. Here are the commands that I did:

git clone git@github.com:username/repo.git cd repo git checkout branch

But that gets me the following error:

fatal: Not a git repository (or any of the parent directories): .git

I do "git branch" and here's what I see:

* default

Where's "branch"?

And let's say I wanted to create my own branch based on "branch". Let's say "branch-zelnaga". How would I do that? Do I just checkout that branch, create a new branch while the current working directory contains files from the desired branch and then push / commit as appropriate?