Madrid to Segovia is easy, and when you get to the train station, you won’t have any trouble finding the bus, because many, if not most, of the people getting off at this station will be taking the bus to town.
This articlesays that the train will stop at Segovia, Valladolid, Palencia and León. After Pola de Lena the train will share the tracks with Cercanías and will stop in Mieres, Oviedo and Gijón. So that means that you should be able to find a high speed train from Segovia to Oviedo.
When you search on Renfe.com (the national rail website), you will see what trains are offered on which routes. In addition to the AVE, the fastest, there are other fast trains, Alvia and Avant. But all you need to know are your starting and ending point, and the website will tell you what your options are.
Not sure if you mean a full day in Segovia and a full day in Oviedo, or if you mean you would travel in to Segovia on one day and then leave the next. I personally don’t think that’s enough time to see either place, and you might want to choose one and skip the other on this trip, to have a longer time in one of the places.