Trains in Spain are excellent, particularly among major cities. You'll take the AVE (alta velocidad = high speed) train from Sevilla to Madrid in 2 1/2 hours or a bit over, and then from Madrid to Barcelona in 2 1/2-3 hours depending on stops. There's also a high-speed train from Madrid to Toledo in under 30 minutes (it's an hour by car). Not every city pair has high-speed train connections, but most of the majors.
I'd probably do 3-4-4 in terms of number of days, but could also see just 2 days in Sevilla if not going anywhere else nearby as a day trip. For day trips, I would do Toledo (am biased as I spent a semester there in college, but it's a gorgeous former capital of Spain) and from Barcelona either Girona or Tarragona or Cadaques/Costa Brava, depending on preferences. Barcelona and the whole region of Cataluņa is amazing for food - I'd try to do one of the top high-end ones if that fits your interests. Celler de Can Roca around Girona is probably impossible to get into, but Disfrutar in Barcelona or sister restaurant Compartir in Cadaques, Tickets in Barcelona, Moments in the Mandarin Oriental or one of several other top ones. There are many, many wonderful places to eat
... You could also do a winery tour. There are also plenty of good restaurants in Madrid and all over Spain.
You might want to check out a football game, either Real Madrid or FC Barcelona (or Atletico, but most people pick one of the first two).
Hope those help with some ideas.