Oh, Kirk. Please don't do this. By the time you get yourself to the train or bus station, wait for departure, ride to your destination, get you luggage to your hotel and get oriented, half the day is gone. You're obviously an experienced traveler, so you know this. What is the point of spending half a day in a magnificent city like Toledo or Cordoba? For pity's sake, scale back the geographical area you plan to cover. Not only are you not going to get much out of the super-short visits to many of your destinations, you are skipping right over some other great places along the way.
OK, I've taken a deep breath. Here are a few thoughts on some of your specific destinations:
Segovia: I really liked it but will admit that it is closer to a one-night destination than many of the other places on your list. I think the bus may be faster than the train, though. Do allow time for wandering; it has much more than just the aqueduct.
Salamanca is a handsome city that somehow wasn't one of my favorites, but it's certainly worthwhile. The historic area is quite large, the train or bus time from Segovia is about 3-1/2 hours, and there's quite a walk from the train or bus station to the core of the historic area. If you can't spend more than one night there, I'd consider skipping it, especially if you don't have a special interest in seeing the university. (I was underwhelmed by my visit to the university--sans guide, but the city is physically attractive and lively due to the student population.)
Santiago de Compostela is another city with a really large historic district, plus a number of museums and sights you'd probably want to visit. Galicia is hard to get to. It will take something in the 7-hour range from Salamanca. Check the historical weather data to be sure you want to spend January time here (and more time getting here). Even in mid-summer Santiago gets a lot of rain and not an awful lot of sun. There are other interesting places to see in Galicia. It seems a shame to travel so far for barely more than one day in the area.
Oviedo has a nice old town but I think its main claim to fame is its very early (9th century?) churches. They are not centrally located and are scattered. Realistically, I think a lunch stop here will be just that, not much of a sightseeing opportunity. If you do stop, there's a fabulous bakery/pastry shop sort of across the street from the train station. Rome2Rio.com gives a train/bus-combo time of almost six hours for the trip from SdC. Just in time for the late Spanish lunch period!
The trip to Bilbao will be by bus. Rome2Rio says 3-1/2 hours. (I haven't independently verified any of these times, which you need to do; Rome2Rio is not terribly reliable.) You don't need a car to get back and forth between Bilbao and San Sebastian; there's very frequent bus service taking less than 1-1/2 hours. A car would speed things up for other parts of your trip, but I've never rented a car in Europe and can't provide any tips on driving in Spain.
I liked both Bilbao and San Sebastian. SS is more overtly a foodie city and at least in summer has a very active tapas culture, but both places have attractive historic districts (healthy walks from their respective bus stations) and one or two important museums. As in Galicia, you should expect cool, wet weather here.
SS-Olite is faster by bus than by train. Olite is relatively non-touristy; might feel pretty dead in January. Interesting to wander the historic district, but you need this time for your other destinations and should skip it.
Besalu is very small and not an overnighting sort of place. Very picturesque, but to me it's leagues less interesting than Girona, which is easier to get to. A day-trip to Besalu is not a good use of your time on such a cramped trip.
I haven't been to Andalucía recently so will leave that area to others. Given the time of year, I'd do Barc - Madrid - Andalucía.
