Your specific requirements will depend largely on the size of your content and volume of traffic, but on average, most shared or dedicated hosting providers will be more than adequate for an OJS install.
To ensure that your error logs only reflect actual errors, at this time it is still best to disable PHP Strict and PHP Deprecated notifications, though this will become less needed over time.
To ensure you are setting up permissions correctly, please review this thread:
It is critical to understand file permissioning if you are hosting OJS yourself.