When I've been to La Rioja, I've traveled to the bodegas with my friends in our rented car, so you may want to do the same. If you choose to travel this way, keep in mind that some bodegas are only open by appointment, so you should contact them first.
If you can read Spanish, this is a web site that names more than 40 bodegas in
La Rioja Alavesa. It's supposed to be in English, too, but the English version wasn't working on my computer.
This tour company called
Cellar Tours was mentioned in my Moon Handbook guide on Spain.
If you want to learn about wine and do a day trip to wineries outside Madrid, this company named
Planeta Vino may be an option for you:
This was yet another suggested company in my Moon Handbook:
Tenedor Tours Vibrant Rioja is the publicity arm of Rioja wines in the United States. They may have some resources for you.
Lastly, there is a website/blog called
Catavino dedicated to Spanish and Portuguese wines that may also have some information for you.
Enjoy La Rioja! The region is beautiful and the food and wine are wonderful!