How would you plan only 2 days in Seville, and then 2 days in Granada?

It's been several years since I've been in Sevilla, but we had a wonderful guide there and I'm pretty sure her name was Conception and her website was Sevilla Walking Tours. We did her tour early in our visit and she explained so much about Sevilla's history and local customs and showed us some great local tapas restaurants for the rest of our days. We liked her city walking tour so much that we signed on for her Alcazar tour, which I actually preferred to the Alhambra. Sevilla's also a good place to see a flamenco show. I had low expectations, but we went to a small venue that really didn't feel too touristy (although I'm sure we all were) and it was a very good evening.
In Granada we found a walking tour that was organized by the city's tourist bureau. In hindsight, I think we might have gotten more out of our visit to the Alhambra if we had had a guide. Choose a hotel in the city center if you're interested in the action which goes on all night!

Hotel Simon in Sevilla was wonderful when we stayed - very central and great value.
http://www.hotelsimonsevilla.com/en_index.html

Main sights are the Alcazar and Cathedral in Sevilla, and Alhambra and Cathedral in Granada.

The above advice is good. I just wonder why you are renting a car. You won't need it in either city and it's probably easier to take the train from one to the other than to drive, especially if you don't want to see places on the way.

Remember to reserve ahead for entry to Granada's Alhambra: http://www.alhambra-tickets.es/. Direct trains run 4x/day from Sevilla to Granada and take 3.25 hours. If those schedules don't work for you, then bus provides a few more options.

I think in terms of nights, rather than days. A day is either a travel day or a tour day. Travel days mean getting out of your lodging, getting to your next destination, checking in and getting settled - which normally leaves little time to tour.

One tour day means two nights, two days means three nights, etc. with a travel day between each destination.

I could do two nights (one full day) in Seville pretty easily. I'd limit my tour day to seeing the Alcazar, Cathedral and strolling the surrounding quarter and down to the river to see the Golden Tower. During my nights, I'd plan on a flamenco performance. If it interests you, there may also be a corrida (bull fight) available as an evening activity.

I think Granada needs two full days (three nights). One full day is for the Alhambra. It's too big for a half day so make that you're one sight for the day and don't hurry through it. You probably won't have seen an Arab bath anywhere else so be sure to find the one in Alhambra (not one of the highly publicized sights), they were an important part of the culture. The other day can be used to see a full variety of sights in Granada based on your interests. I would plan to see a sambra dance performance in one of the cave homes on the far side of the Albaycin. It's like flamenco but still unique.

The "must" for Granada is the Alhambra, and as many people mentioned, you HAVE to buy your tickets in advance or you won't get int without doing a prepaid tour. It may already be too late to get tickets for June, but buy them now!!!