I'm used to PHP style MVC frameworks which are a little different from the way ASP's MVC frameworks work.
Is there a built in database class? Or do I need to code my own database class using a web.conf connection string?
With PHP's MVC frameworks you simply add your database connection information into a database and then query it with something like:
$results = $this->db->query("select * from table");
It is not quite that simple, but you have a lot more flexibility. Check this link out: http://www.dotnetperls.com/sqlconnection -- It has some getting started for doing exactly what you've outlined (though I HIGHLY recommend you use sprocs or at a minimum, paramaterized sql, not direct sql).
With that said, I suggest you look into Linq-To-SQL or Entity Framework. Any on-line sampels you find will probably use one of these technologies, and they do make things very nice and simple.
Use an ORM like Entity Framework 4.1 or nHibernate. They will handle the database connection internally and therefore abstract away some of the complexities of safely opening/closing connections. Then all you need to provide is a connection string.
