You can get 500 errors if you have a mistake in your .htaccess. Switch to default permalinks (if you can) and either delete or rename your .htaccess to something else. Then check out:
http://codex.wordpress.org/Using_Permalinks
Yes of curse you install at locally and keep practice on. Once your build your project or task you can move it on your hosting domain. But first one is to install your local system.