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.
That`s great. You must visit Laurel Street for a quick lunch or better dinner. You can go for one bar to another (they are beside one to another), and you can have a little "tapa" and a glass of Rioja wine. There are many people in the street drinking and eating, if the weather allows.Don`t miss it.