In the Old Town there are many churches you should visit. You shouldn't forget the Cathedral La Redonda, the Romanic Church of San Bartolome and the church of Santiago that is visited by pilgrims in their way to Santiago de Compostela.
If you want you to know more about the wine from La Rioja you can visit several bodegas. I recommend Juan Alcorta in Logroño, and you can visit (http://www.lariojaturismo.com/) for further information and other town`s bodegas.