What type of NN should be used for a turn-based game (with direct access to data)?
Your question is quite broad so it is not easy to give a direct answer.  However, a simple approach and a good introduction to machine learning would be to use a Multilayer Perceptron (MLP).  This is one of the most basic and fundamental neural network architectures.

You will also want to read about Backpropagation, which is the standard method of training such algorithms.

Here are some code examples/libraries (in no particular order) that can get you started:


Keras - (Python)
Theano - (Python)
DeepLearning4J - (Java)
Caffe  - (C++/Python/Matlab)
TensorFlow - (C++/Python)
Weka - (Java)