Train from Cordoba to Toledo

Not on a train. Trains to Toledo go through Madrid as a rule (there may be exceptions, but I haven't seen them and Cordoba isn't one).

I think the same is true for busses, but I'm not sure. Regardless, the train is high speed so even with a change will be faster than a bus.

Trains to Toledo can ONLY be reached by the line directly from Atocha Station in Madrid. It is essentially a commuter line with trains about every 30-45 minutes. From Cordoba, you can get a a train to Madrid (Puerta de Atocha Station). Once arrived, you can buy a ticket for the next train to Toledo.

When you buy your train tickets to Toledo, do consider buying the round trip ticket, especially if you intend to return to Madrid in the afternoon or evening, as these tickets may sell out.

Since you plan to spend two nights in Toledo, you can leave part of your luggage in a secure locker at the Atocha station, if you wish.
When my mom & I visited Toledo, we left one of our carry-on roller bags in the locker and only took one with us; it was so much easier to deal with one instead of 2 roller suitcases! ;-)

Enjoy your trip!

As others have said, it isn´t possible to go directly from Córdoba to Toledo by train. However, it is not at all difficult to change trains in Madrid. I have done that very trip myself in the reverse (from Toledo to Córdoba). It was very simple. The trains are super fast and very comfortable.

The rail system in Spain is not as extensive as other parts of Europe. Therefore, Spain has a very extensive bus system that is cheap and pretty fast. I am sure you can get a bus to Toledo from Cordoba. The bus station in Cordoba is across the street from the train station. We frequently use the buses when in Spain.

I think a bus will take longer than the train on this route. Although you have to double-back on the train, the Cordoba-Madrid leg is a high-speed train moving much faster than a bus.