As your itinerary seems pretty open right now, here is a suggestion. I am thinking of 9 nights and a car rental, flying in and out of Barcelona.
Nights 1,2,3 and 4 in Barcelona.
Day 5 AVE to Zaragoza and Night in Zaragoza,
Day 6 rent a car, visit the Monastery of Veruela and the Borja DO museum.
Night 6, 7 and 8 in Logroño.
Day 9 back to Zaragoza, drop the car off and train back to Barcelona. Night 9 in Barcelona.
A few comments. Zaragoza is worth a visit. It is an underrated city, and most people find it interesting.
The Monastery of Veruela is beautiful. There is a lot of talk about overtourism, and how many places are full of crowds. I have visited there several times, and there has never been more than 5 or 6 people visiting. The D.O. of Borja has it´s museum/interpretation center there and you can taste their wines.
If you are in La Rioja for more than two nights I recommend Logroño. For one or two nights a place like Briones is okay, but not a lot of food options.
You don´t say when you are visiting. Depending on the time of yerr there are different fairs and fiestas. Mid-September is harvest time and San Mateos, so La Rioja gets a bit hectic. My favorite time is the last half of October, there is still plenty of light, the weather is good, and there is still plenty of color in the vineyards.