Take bus 69 from the Delft railway station, direction Rotterdam, and get off at the last stop, about 3 km from the railway station. The stop is just to the south of the grounds of the Reactor Institute. Alternatively, bus 40, stop "Kluyverweg". The reactor is one block to the east of the stop, at the corner of the Kluyverweg and the Mekelweg. For more information see de Reiswijzer (in Dutch).
Here is a map to use.
Take the train from Schiphol airport in the direction Den Haag or Rotterdam. You will probably have to change trains in Leiden. The train to Delft leaves from platform 9 in Leiden. The whole trip by train takes less than 1 hour. In rush hour traffic, a taxi from the airport may take considerably longer. A one-way train ticket costs about 10 euro, a taxi from the airport 80 to 100 euro.
To find persons of the Department of Radiation Science and Technology you can search the university phonebook.