Traveling for 3 weeks in Spain and Italy

Arrive in Barcelona - 4N
Fly to Granada - 2N
(Vueling offer direct flights)
Train to Sevilla - 4N
(with day to Cordoba)
Train to Madrid - 4N
(with day to Toledo)
Fly from Madrid to Florence - 4N
(Vueling offer some direct flights)
(with day to Siena)
Train to Rome - 3N
Depart from Rome

What are your travel dates? And are you flying into Spain from the US or Canada?

The first thing I'd do is go to skyscanner.com and check on the city pairs with non-stop flights on the day(s) you hope to make the country switch. Your trip is not a long one for visiting two such large and diverse countries, and you can't really afford to waste time with a long trek through Spain to get to your departure airport or through Italy from your arrival airport to your first Italian city. And with just a week in Italy, you'll probably want to limit yourself to destinations that are reasonably short train rides from Rome (don't worry, that includes Florence and Venice).

Edited to add: David's itinerary works for me, though I'd have a hard time with such a short stay in Barcelona, especially given that your first day will probably be jet-lagged if your flight is transatlantic.

"Any suggested itineraries?"

Sure, hundreds. But to give you advice, we need to know a lot more. For example, have you been to either of these places before? What made you choose these countries, and this particular 2/3 vs 1/3 time allotment? What time of year are you going? Do you have specific interests (car museums, movie locations, olive oil, mountain views)? Do fine arts museums thrill you or bore you? Are you wanting to rent a car, wanting to avoid car rental, or open to either option? Are you traveling alone or with others? If others, what are their interests - and their ages (any children or adolescents)?

Etc, etc. Help us help you!

If you need more help narrowing some of this down, Rick's Spain and Italy books will have his suggested highlights of each country, and if you haven't been before, these are excellent places to start. You can also look at his videos, here, which will give you enough ideas for a trip of 3 months rather than 3 weeks. Click here, then scroll down to choose Italy and Spain: https://www.ricksteves.com/watch-read-listen/video/tv-show

I have not been to either of these places and I am honestly not set on this particular time allotment. I am flexible. I am traveling in May. I enjoy nature, hiking, yoga, trying new food, good drinks, learning about other cultures, and I appreciate art. I do not want to rent a car, I hope to travel by train most of the time. Also I am traveling alone for majority of the trip and meeting some friends for the last week.

May is the perfect time to enjoy nature, hiking and yoga.
Any time is a great time for learning about other cultures, and appreciating art.
All the time is the best time for trying new food and good drinks.

I flew from Bologna to Sevilla on a budget airline. Both are small airports so you don't need as much time.

Barcelona and Andalucia (skip Madrid unless you want to see the art museums) then from Bologna there's plenty to see in Tuscany and/or Umbria "on the way" to Rome.

Make Cordoba a stop on your way from Seville to Madrid, that saves time backtracking. Eat and sleep better and cheaper in Toledo (a much nicer place IMO than Madrid).