A direct flight connects Madrid with Logroño. You can book it at Iberia website. We have also made a cool map with the nearest airports and the time it gets to come to Logroño, check it out!
A four hours from Madrid or six hours from Barcelona travel, you could also arrive to Logroño by bus. Alsa is the largest bus company of Spain.
Several trains arrive to Logroño from the whole country. More information is available at Renfe website.
The easiest way is to plan your trip using Google Maps :)