If you only have one day, focus on the old town. This is a planned route; personal notes will be added as it is walked and refined.
The route
Morning: start at the Mezquita-Cathedral, before the day heats up. Midday: walk through the Judería and pause for lunch. Afternoon: cross the Roman Bridge towards the Calahorra Tower for views back over the river, then visit the Alcázar de los Reyes Cristianos. Evening: tapas in the old town.
Adjust the order to the opening times you find on official sites, and keep the hottest hours for indoor stops in summer.
Keep exploring
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