Accessing + retrieving custom database in WordPress

It is OK. And you can access this table with standard way, throught wpdb

To insert data to your table use this code:

global $wpdb;
$wpdb->insert( 
    'exampleTable', 
    array( 
        'column1' => 'value1', 
        'column2' => 123 
    ), 
    array( 
        '%s', 
        '%d' 
    ) 
);

To breafly explain what this code mean. WordPress sanitizes values before inserting them into database (to prevent database being compromited by SQL injection.). First line defines table to inser value. Secon array contains col names and value being inserted. The third array contains type of value to insers. %s stands for string, %d stands for int (digit).

You can read more about $wpdb->prepare in the codex. It is well explained in there.