The trouble I'm foreseeing with this is that every instance of every model will open and maintain its own MySQL connection, and this all happens on a per-request basis. So if 10 people visit a page that calls 10 objects you've just occupied 100/250 connections, 250 being the default max on a fresh MySQL install.

You want things like this to scale in a linear way, not geometrically.

Instead, instantiate a single DB connection and pass that to your models. [Dependency Injection]

Secondly, I would offload calls to things like prepare() and execute() to the Connection object, as that is where you would want to handle related tasks like error handling. eg:

class Connection {
    function query($sql, $params=NULL) {
        if( ! $sth = $this->dbh->prepare($sql) ) {
            $err_arr = $this->dbh->errorInfo();
            $err_msg = sprintf("SQLSTATE ERR: %s<br />\nmySQL ERR: %s<br />\nMessage: %s<br />\n", $err_arr[0], $err_arr[1], $err_arr[2]);
            throw new Exception($err_msg);
        }
        if( ! $sth->execute($params) ) {
            $err_arr = $sth->errorInfo();
            $err_msg = sprintf("SQLSTATE ERR: %s<br />\nmySQL ERR: %s<br />\nMessage: %s<br />\n", $err_arr[0], $err_arr[1], $err_arr[2]);
            throw new Exception($err_msg);
        }
        return $sth->fetchAll(PDO::FETCH_ASSOC);
    }
}

$result = $conn->query(
    'SELECT foo FROM bar WHERE baz = :param OR bla = :param',
     array('param' => 'shoop'));

The trouble I'm foreseeing with this is that every instance of every model will open and maintain its own MySQL connection, and this all happens on a per-request basis. So if 10 people visit a page that calls 10 objects you've just occupied 100/250 connections, 250 being the default max on a fresh MySQL install.

You want things like this to scale in a linear way, not geometrically.

Instead, instantiate a single DB connection and pass that to your models. [Dependency Injection]

Secondly, I would offload calls to things like prepare() and execute() to the Connection object, as that is where you would want to handle related tasks like error handling. eg:

class Connection {
    function query($sql, $params=NULL) {
        if( ! $sth = $this->dbh->prepare($sql) ) {
            $err_arr = $this->dbh->errorInfo();
            $err_msg = sprintf("SQLSTATE ERR: %s<br />\nmySQL ERR: %s<br />\nMessage: %s<br />\n", $err_arr[0], $err_arr[1], $err_arr[2]);
            throw new Exception($err_msg);
        }
        if( ! $sth->execute($params) ) {
            $err_arr = $sth->errorInfo();
            $err_msg = sprintf("SQLSTATE ERR: %s<br />\nmySQL ERR: %s<br />\nMessage: %s<br />\n", $err_arr[0], $err_arr[1], $err_arr[2]);
            throw new Exception($err_msg);
        }
        return $sth->fetchAll(PDO::FETCH_ASSOC);
    }
}

class Model {
    function __construct($conn) {
        $this->conn = $conn;
    }
    function insert() {
        $sql = "INSERT INTO foo (bar, baz) VALUES (:id, :value)";
        $this->conn->query($sql, array(
            'id' => $this->id,
            'value' => $this->value));
    }
}