What do you want to use the network for?
If you want to test it on mnist, you can run caffe test [...] to score the network on the mnist dataset (that's using the compiled C++ binary).
If you want to use it on a different dataset with different classes, you can finetune the network using caffe train -weights=<model_binary_name>.caffemodel [...], and test it on that dataset using the above.
If you want to use it as a generic feature extractor, you can use the extract_features C++ tool.
I am guessing you can also do all these with the python and matlab wrappers, so which language you use is purely a personal choice.