Choosing a learning rate
Is the learning rate related to the shape of the error gradient, as
it dictates the rate of descent?


In plain SGD, the answer is no. A global learning rate is used which is indifferent to the error gradient. However, the intuition you are getting at has inspired various modifications of the SGD update rule.

If so, how do you use this information to inform your decision about a value?


Adagrad is the most widely known of these and scales a global learning rate η on each dimension based on l2 norm of the history of the error gradient gt on each dimension:


Adadelta is another such training algorithm which uses both the error gradient history like adagrad and the weight update history and has the advantage of not having to set a learning rate at all.

If it's not what sort of values should I choose, and how should I choose them?


Setting learning rates for plain SGD in neural nets is usually a
process of starting with a sane value such as 0.01 and then doing cross-validation
to find an optimal value. Typical values range over a few orders of
magnitude from 0.0001 up to 1.

It seems like you would want small values to avoid overshooting, but
how do you choose one such that you don't get stuck in local minima
or take too long to descend? Does it make sense to have a constant learning rate, or should I use some metric to alter its value as I get nearer a minimum in the gradient?


Usually, the value that's best is near the highest stable learning
rate and learning rate decay/annealing (either linear or
exponentially) is used over the course of training. The reason behind this is that early on there is a clear learning signal so aggressive updates encourage exploration while later on the smaller learning rates allow for more delicate exploitation of local error surface.