I wouldn't do it that way at all, it'll be full of security holes if you're accessing the database directly from the client.
However, mysql (www.mysql.com) would be the immediate replacement for sqllite. There are other options, but mysql is a reasonable first step up from sqllite.
As I said, I wouldn't recommend doing it this way. Instead write a simple server application that performs the necessary database queries in response to messages from the client, so that people don't have direct access to the database.