Continuing the idea that working with bare presentations is "hard", automatic groups sometimes can give you a handle, if you're trying to study a specific group . Automatic groups are groups with finite state machines that can, essentially, solve the word problem for that group.

If you're studying a particular group and are lucky, a procedure such as Knuth-Bendix can compute an automatic structure for you. Then lots of hard computations become easy (e.g. the order of the group).

Magma has some of these algorithms implemented, see this Magma documentation page.