Valladolid airport has connection with Barcelona airport.
By bus: the company Alsa connects with Valladolid from the airport. Departures from terminal T4, floor 0, Arrivals area. Bus Company Website
By train: the train arrives at the “T4 Airport” station, which is located at the airport, in the T4 terminal -1 plant. It connects with Chamartín Station where you can take an AVE (high speed) train or long-distance train to Valladolid. Railway Company Website
From Madrid-Atocha Train Station you can access to taxi, subway or train to Chamartín Station and take an AVE train (high speed) or long-distance train to Valladolid.
Barcelona – El Prat airport has connection to Valladolid by plane, bus or train. The airport also has a subway and taxi service and car rental.
By plane: Check flights. Barcelona Airport Flights
By bus: From the T1 and T2 terminals you can travel to Metropolitan area of Barcelona with the regular buses lines TMB. From the Barcelona Nord Bus Station you can travel to Madrid-South Bus Station. In both cases, from there you can also travel by bus to Valladolid.
By train: R2 line of Rodalies (Cercanías Renfe) is currently the train of Barcelona – El Prat airport (T2B) and stops in Barcelona – Sants, where the Train Station is located, from which you can take the AVE train to Madrid -Atocha every 3 hours and the “Combined” train every 6 hours. From Madrid-Atocha Train Station you can access to taxi, subway or train to Chamartín Station and take an AVE train (high speed) or long-distance train to Valladolid.
You can check here whether you require a visa to enter Spain.