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If you have a recent version of phpMyAdmin, you can configure it (in its config.ini.php file) to export and import using local files, rather than uploading and downloading.
I had some difficulty figuring out how to specify the right paths for that, but here's a working example that I recently used:
/**
* Webserver upload/save/import directories
*/
$cfg['UploadDir'] = './upload';
$cfg['SaveDir'] = './download';
This helps with larger databases, but isn't always adequate if you're on a slow server. Sometimes you have to backup and restore the database in pieces: individual tables, or even pieces of individual tables, which is a big pain to do.
Being able to use the mysqldump and MySQL commands from the shell, via SSH, is the best way, if that's available to you.