~14 day itinerary in Spain in Feb

I am currently in Madrid for the 2nd time and my perspective has changed for the better. It is a nice winter destination. This is what I recommend
Fly into Madrid Train to Toledo
Toledo 2 nights
Train to Madrid
Madrid 5 nights (day trip to Segovia)
Train to Barcelona
Barcelona 7 nights (include day trips)
Fly home

If you have not already looked at the RS tour itinerary, it may provide some info/inspiration

https://www.ricksteves.com/tours/spain-portugal/spain

The AVE trains between Barcelona and Madrid are extremely fast--faster than flying by the time you make the treks to and from the airports. There's no reason to consider any other form of transportation between those two cities.

Not sure on your dates, but Barcelona is expensive at the end of February due to a large convention.

This is another RS resource for info on Spain.

And if you haven't been to Europe before, thoroughly read and take to heart all the information presented in the Travel Tips section.

As jleonardi says, you may want to also check the reverse itenerary, into Barcelona & out of Madrid, both hotel and flight prices.

For me, Barcelona and Andalucia (Sevilla, Cordoba, Granada . . . ) are more interesting than Madrid. Unless you really love European painting (Madrid has 3 superb art museums), I'd advise you to go straight to Andalucia, then Barcelona - or vice versa.

I agree with Chani. Skip Madrid and head to Andalusia, which would be lovely in February.

Fly to Sevilla (3 nights)
Rent a car and explore Pueblos Blancos (1 night in Ronda, Arcos, Grazalema, or take your pick)
Cordoba (2 nights)
Granada (3 nights)
Fly to Barcelona (5 nights)

More variety and locations in a warmer part of the world than Madrid in February.

I REALLY agree with Chani. Additionally, at that time of year, the weather should be quite lovely in Andalusia. I just took a approx 13 day trip to Spain in November. I'd already been to Barcelona. Madrid is nice enough, but in a country with wonderful history and cities, Madrid would be low on a list. Our itinerary was as follows:
1. Arrived Madrid airport and took the train to Sevilla for 3 nights.
2. Rented a car in Sevilla and spent 2 nights in the Pueblos Blancos area
3. Drove to Granada and spent 2 nights.
4. Train to Cordoba and spent 2 nights
5. Train to Toledo (via Madrid) stayed 1 night
6. Train to Madrid and spent 2 nights with a day trip to Segovia

Nathan's idea is very good, but you will want to visit both Sevilla and Cordoba by train, then rent a car through the pueblos blancos and return it when you get to Granada. If you can fly into Sevilla, start there. If flying into Madrid works better, then Cordoba first, just because it's on the way to Sevilla. Maybe 2N in Ronda or 1N in another town and only 2N in Granada.

If you really want to see the art museums of Madrid, start there with 2N and drop the pueblos blancos.

I've been to Spain twice in February for 3+ weeks. The weather's been mostly comfortable, high temps in mid-50s to low 60s and much sun. Madrid's weather was very similar to Andalucia. Be prepared for some rain and grey skies, but also for sitting in the sun and enjoying iced drinks