Cities in Spain to visit

Barcelona 3 nights
Train to Cordoba
Cordoba 2 nights
Train to Seville
Seville 3 nights
Train or Bus to Granada
Granada 3 nights
Train to Madrid
Madrid 2 nights

I did something similar (except for Barcelona where I've been 2 times) about a month ago

Fly in to Barcelona
Barcelona (4 nights) - Day trip to Montserrat Monastery
Fly low-cost carrier to
Granada (2 nights)
ALSA Bus to
Sevilla (3 nights) - Day trip to Jerez
AVE Train to
Córdoba (2 nights)
AVE Train to
Madrid (2 nights) - Day trip to Toledo
Fly out of Madrid

I recommend flying directly from Barcelona to Granada (1.5 hr), using a low cost carrier, so that you don't have to backtrack all the way back to Madrid. It also saves you a longish train journey down to Andalucía (more than 5 hours even with the AVE Train).

I also gave Madrid only 2 nights since you indicate you are not interested in museums.

May I ask when you plan on traveling to Spain? (I hope it's not in August🔥!)

For routing, fly Barcelona to Granada (it is a long train/bus route otherwise) - Seville - Cordoba - Madrid.
I was in Spain Oct/Nov 2017 at which time it was 'imminent' that the train would be running directly from Granada to Seville. I happened to just check a post in Dec 2018 on Tripadvisor, and it is still not connected. Therefore, just take the Alsa bus from Granada to Seville. Time is the same as the train bandaid route and cheaper.
Some people do Cordoba as a day trip from Seville, some do it as a day trip enroute to Madrid, others spend an overnight or more to get a better feel for the place.
Even with 14 days, depending on arrival and departure - this could be too many stops, but perhaps 4-2-4-4. Keep in mind some people like to count nights in a place instead of days in a place, and it really depends on what you want to do. 4 nights in Granada was just about right for me.
One consideration is that Monday closures of sites can mess up short stay plans.
I use an excel spreadsheet, list the dates and days of the week, enter time of arrival and departure. I have a column for activities, and a column of where I am sleeping. Sometimes, I move to the next city in the morning, other times, I do it late in the afternoon so I can get a little more time in the first city. Listing the date and days means you can see Monday and work around it. You also want to keep in mind there are a lot of local feast and holidays we don't think of, so be sure you know what is going on when you want to travel. It might not affect closures, but it might mean crowds when the locals pour into town, or the prices and availablility of lodgings. For example, Seville has flamenco events after Easter which can really mess up tourist plans. However, if the Fall, then you need to look for other events.

I think Carlos has given you the perfect itinerary.

I will be going in late September or early October

@quansoon Ok no worries then! :)
I'm not sure when you plan to be in Barcelona specifically, but FYI you may be in Barcelona for our annual city festival, La Mercè (Sep 21-24). The festival itself is huge, there are 100s of events spread throughout Barcelona's squares, streets, museums, and parks. The best part - all the entertainment is free! This is something that we put on for ourselves, not just for tourists. If you want to experience Catalan culture like a local, at least for a few days, this is a great opportunity.

I would recommend you shoot for at least a couple of days in Barcelona during the Mercè festivities. Here is the festival program from last year: https://www.barcelona.cat/lamerce/en/programa-per-tematica
I would keep an eye out on this website (the official one) for when they put up the 2019 program.

To modify Carlos’ Itinerary a bit for 14 days:

Fly into Barcelona

Barcelona (4 nights) - Day trip to Montserrat Monastery or Girona

Fly low-cost carrier (Easyjet, Ryan Air, or Wizzair) to

Granada (2 nights)

ALSA Bus to

Seville (4 nights) - Day trip to Jerez & Cordoba

AVE Train to

Madrid (4 nights) - Day trip to Toledo & El Escorial

Fly out of Madrid

Thank you this has been very helpful