Travel in Spain

If three of us are going to Madrid, then Granada and ending in Seville (one week), should we rent a car or take public transportation?

It would be foolish to rent a car for that journey. You'll curse that car once you try to drive out of Madrid and/or try to drive into Seville. And that's before you try looking for somewhere to park in Seville. Use public transport. BTW, is the 'one week' for the whole trip? If so, I'd advise you to think again - far too ambitious. Two of your days are going to be spent on getting from A to B.

Public transport is really easy in Spain. Sit back and relax :-)

Car renters isolate themselves from one of the best parts of the trip. Trains let you have long conversations with helpful interesting people. We learn all kinds of good stuff by accident.

You won´t use a car in cities anyway. Meanwhile you´re stressed, vulnerable, and charged for insurance, at hi cost, while you´re not using it.
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Why not go home a little wiser than when you left?

The only part of the trip where I can see that a car would come in handy, is if you decide to explore Andalusia. Although there is public transport it would make life a little bit easier. But this is only true for visiting the small towns. In the cities you would find any use for it. (I agree with gawkabout, on that). For Madrid get a 10 viajes ticket for the Metro. Granada and Sevilla you can do by foot.

There are enough flights at a reasonable price to get you from Madrid to Granada (Check Iberia and Spanair, a bus is standing by to take you to Granada after landing). The trainconnection between Granada and Sevilla is perfect. (4x daily, 3 hours), the trip back to Madrid from Sevilla can be costly if you take the AVE (Spanish TGV), and I would recommend a reservation for the AVE. But in Spain you can always take the bus.

Don´t rent a car, there are cheap busses between this cities, check "sevibus" for madrid- granada /sevilla and "Alsina graells" for Granada/Sevilla