Homebrew warning: "No developer tools installed" even though Xcode is installed

After installing Homebrew the "brew doctor" command returns:

Warning: No developer tools installed.
You should install the Command Line Tools.
Run `xcode-select --install` to install them.

…and I do have Xcode installed.

Anyway, after running xcode-select --install the Xcode installer appears, then I click "install" and the installer closes.

Is this normal? …and do I have to install Git from Homebrew, or can I use the one that comes with Xcode?

Yes, it is perfectly normal for homebrew to prompt you to install Xcode. That happened to me right after I installed Mavericks.

As for your question about git, it depends a lot on your usage. Default git (and almost every other tools) are not of the latest version. If you're OK with that, then of course you can just use the default one.

Xcode itself isn't what i needed to install, it was Specifically the "Xcode Command Line Tools", apparently thats a different thing...