Itinerary suggestions

Fly into Madrid
Madrid 3 nights
Train to Granada
Granada 3 nights
Fly to Barcelona
Barcelona 4 nights
Fly home from Barcelona

You'll need to stay on top of the ticket situation for the Alhambra. Tickets can sell out months in advance. Here's the official ticket website.

With only ten days (how many nights in Spain, not counting the night on the train [I meant plane, of course]) you cannot see all the top destinations in Spain. You need to decide what destinations are most important for you. One thing to keep in mind is that Holy Week is a very big deal in Andalucía. Hotel rates rise at that time, the major cities in the south are rather crowded, and in some cases there are large processions and adjustments to the hours of operation of some of the major tourist sights.

Reasonable plan, but consider starting Madrid-Barcelona by the fast train.

With 10 days (11 nights?), I think you could have an interesting and varied holiday stopping in just Madrid and Barcelona, travelling between them by railway, and taking a few day-trips from each city.

Alternatively, if you want to see some of Moorish Spain, you could travel between Madrid and Barcelona with a break of a night or two in Zaragoza (aka Saragossa), again travelling between the three cities by rail.

If you want to include the South, however, I'd drop one of either Madrid or Barcelona (I'm assuming these are your entry/exit points). For example, Madrid - Sevilla - Granada - Madrid Airport by rail/coach.

I recommend spending a night or two in Cordoba. If it were my trip, I would do Madrid, Seville, Granada, Cordoba and back to Madrid. Or, better still, consider an open-jaw or multi-city ticket, if the price isn't outrageous. Such as fly into Madrid and fly home from Seville. This way you don't waste time backtracking and you don't waste money returning to Madrid. I would save Barcelona for another trip to Spain. Barcelona is a large city and has lots of interesting sites to see, many of which now require buying tickets in advance. To see Barcelona properly, I think you really need 5 or 6 nights, which you don't have on this trip if you include the other destinations.

Spring is a nice time to visit Andalucia because the temps are still comfortable. Summer months are extremely hot in Andalucia.

Add one night to Madrid, so you can do day trips to Toledo and Segovia. 2 nights for Granada. Why not see Seville instead of Barcelona this time.