cheap and expedient

I have been to all the towns you are trying to reach but did it two separate trips to Spain.
Hence, you are wanting to go to too many place in the amount of nights you have earmarked.
On this trip you have to either forget Toledo and Segovia or Granada and Seville.
I would either fly from Gibraltar to Madrid and use that as a base to see Toledo and Segovia

or I would just take the bus to Granada or Seville from Gibraltar and spend time in either city then train to Madrid

There is no way you can do all of that in the time you have available. At best you may be able to see Granada and possibly either Cordoba or Segovia. I would recommend you look at travel times between cities and then pick one or possibly two cities on your list. You will need to be in Madrid on Tuesday night for a Wednesday morning departure.

Agree with the others: you don't have enough time to do the things you describe.
You will want to spend serious time with a map so that you're familiar with the distances and travel times of the area you've outlined.
Just one example: It will take you most of the day on Tuesday to get to Madrid for your Wed. am flight, you need to arrive there Tuesday night.
And trying to drive to Madrid would involve more mileage and time than you're probably thinking, which suggests train or flying to Madrid.
The Rome2Rio website may save you some time in researching travel times.
Re Granada, entry into the #1 attraction in Granada, the Alhambra, must be booked online several months in advance.

With Sunday and Tuesday being travel days, you really only have one full day to spend at any one of these cities. Let’s say you choose Granada. That’s a 5.5 hour bus/train ride or almost 3 hours driving a rental car. Then on Tuesday, it’s a 3.5 hour train ride (at 7 am, 3 pm or 7 pm) or 5 hour drive to Madrid.

I would recommend that you
1) Decide whether you want cheap OR expedient transportation. Buses are cheap, but not necessarily expedient.

2) Identify what it means to you to “tour” a city. This is very individual. Does it mean: store your bags at the train station, see one site, get back on a train and continue to the next city? Or spend 1-2 days exploring one city?

3) Look at transportation options between Gibraltar, one or two (max) cities that interest you, and Madrid. Then determine the places that work within those options. Plan to sleep in Madrid Tuesday night before your Wednesday flight.

With this short amount of time, I would pick Sevilla for Sunday & Monday, and take the train to Madrid Tuesday. Even that is cutting those towns short of everything to see and do.

A few thoughts...you are arriving in Gibraltar, which is a cruise port. There might be a few local tour companies offering extensions for cruise customers who would like to see some of the areas you mention and then fly home from Madrid. Perhaps the Gibraltar tourism office can help you connect to these.

Otherwise, The only other thing is Road Trip!! A one way car rental from Europcar in Linea- pickup Saturday so you’re ready to hit the road to Granada on Sunday morning. Arrive around 1 pm Granada and take a walking tour or visit Alhambra (book far ahead). Night in Granada.

Monday, drive to Cordoba, spend the afternoon. It’s 220 miles to Toledo so drive on and stay somewhere in the countryside en route. Book ahead.

Tuesday, drive rest of the way into Toledo and see the sites. Drop the car here and take the train into Madrid. Night in Madrid.

I did not suggest bus/train alternative because of the 5+ hour trip from Gibraltar to Granada. Now I like to drive and I’ve driven in Spain, including the South. But while this itinerary is doable if you really are just wanting to sample these places, it is not ideal for a relaxing holiday.

Previous posters have offered some wise advice. My purpose was to lay out an actual itinerary you requested, day by day. Hope this helps.

Will you have already spent any time in Madrid?

Do you have any interest in great art?

Where and how are you going to eat if restaurants aren't in your plans?

But most of all, what time of year are you planning to go? That definitely can affect how much you can see and do wherever you are.

I agree that you need to do a reality check using Rome2rio. It's not perfect, but it will provide some eye-opening info on the kinds of transportation you can use to get from one place to another and how long each of them will take. When you get your results, be sure to click on the little map symbol to see where the locations you listed are.

Typically, going from one place to another takes more time than just the time on the mode of transportation. So 2 nights are recommended in the destination in order to have the full day between them to see and experience some of what's there.

Using that idea, Sunday would be a travel day to a chosen destination, Monday would be a tour day of that place, and Tuesday would be a travel day from there to Madrid so that you can make that flight on Wednesday morning.

Of course, if you go straight to Madrid from Gibralter, you could have a couple of days there to explore it, take a day trip or maybe both.

Even a road trip barely lets you do more than wave at the towns on your way by.
I also suggest using rome2rio for logistics of transport. Personally, I would research flights from Gibralter to Madrid and do some day trips from there. Sevilla, Cordoba and Granada can be left for a longer trip in the future.

I do not find Rome2Rio trustworthy beyond identifying the forms of transportation available (though it can vary by day of the week) and the places you may need to transfer. The fares, travel times and frequencies are often wildly inaccurate. You'll need to keep drilling down ro find the name of the company operating tne trains or buses and go to its website for reliable information. Rome2Rio usually provides a link. If it does not, use renfe.com for train schedules and fares and Google something like Bus Gibraltar to Cordoba for bus info.

ViaMichelin.com probably provides the most accurate driving times, but you must pad them quite a bit to allow for stopping, traffic, getting lost, looking for parking, and getting to your sightseeing targets on foot from wherever you are forced to park.