9 nights...Barcelona/Madrid/Seville

I think you need to search for best open jaw airfare flying from your home city
wherever that is to one of the 3 cities and then from one of the remaining 2 cities back home. Then you plan accordingly. For me, I found flying to BCN and home from MAD was cheapest and fastest. So if it were me, in between, I would fly from BCN to Seville (to save time and avoid long train) and take a fast train to MAD before fly home.

FWIW, nine nights is pretty short for those major cities. With 9 nights, I would just do 2 cities instead of 3. I would want at least 14 to do those 3 cities. But that is me. Good luck.

Two nights in Seville gives you only one full day and maybe part of the day you arrive and part of the day you leave, depending on your flight and train schedules.

Also, Seville will be very, very hot in mid June. Think of temps in the 90's and 100's. I love Seville. It is beautiful but summer is not the best time to visit unless you have a high tolerance for heat. If you do visit Seville this June, be sure to reserve a hotel with a pool.

With only 9 nights on the ground, I would visit 2 cities. Otherwise, you are spending a lot of time traveling to your destinations, packing/unpacking, checking in and our of hotels.

One possible itinerary is to spend 4 nights in Madrid and 5 nights in Barcelona. You don't mention what your interests are. Madrid has some outstanding art museums, the Royal Palace, Retiro Park, San Miguel market, Temple Debod. Barcelona has tons of quirky Gaudi buildings, interesting churches and cathedrals, the Boqueria market, art museums and interesting neighborhoods. And Barceloneta beach. I think you need more time in Barcelona than in Madrid.

If you really, really want to see Seville for a taste, I would fly into Madrid first and spend 3 nights there, then 2 nights in Seville, then 4 nights in Barcelona. Stay in Madrid when you arrive for your 3 nights. Don't head to Seville immediately and then backtrack to Madrid. You are wasting too much time traveling back and forth.

Have you checked some guidebooks?

You need to carefully pick which museums you want to see and which you are willing to acquiesce to skipping. I don't have experience with Barcelona. I spent 4 nights/ 3 whole days in Seville, (2 nights in granada, 2 nights in cordoba), 5 nights/4 whole days in Madrid but one of those 4 days was a day trip to Toledo, which you don't have time for.

You might want to skip either Barcelona or Seville on this trip.

What do you want to see in Seville in only one to 1-1/2 days in Seville? Don't rush through any museum but see and observe the one or 2 places you pick, good. Many opinions including me, say Seville deserves 4 or 5 nights.

After you check the guidebooks and get a good idea of what you want to see, would you like to post a supposed day- by-day itinerary of the major sites you want to see, transportation, and which cities you are sleeping in which nights and approximate times of day you want to leave or arrive in each city?

Cut one city and do a thorough visit of two, otherwise you will waste time traveling.

Ditto on doing two of the three. I would save Sevilla for a full Andalucia experience.

I did almost this same itinerary in September, 2017. My friends and I flew in to Madrid where we spent four nights (with day trips to Segovia & to Toledo), took the train to Seville for two nights, and then took the train to Barcelona for four nights. Yes, it would have been wonderful to have more time in each location, but we still had a fantastic time and got to see a lot. I say go for it! And if you fall in love with Spain (as I did on that trip), you can always plan on going back and spending more time.