For example casual, polite, and honorific
It seems that what you're asking about are formality levels in Japanese. As some other have noted, they can be broadly classified as "casual," "polite," "humble," and "honorific," but even within these classifications one can make their speech more/less formal depending on the context (e.g. the level formality one would use when speaking to the emperor is much higher than when one is speaking to the president of their company).
Are these the only different ways or are there some other ways that an expert speaker would know / use?
An "expert" speaker (whatever we'd like that to mean) mixes and matches various speech levels based on the context in which they are in and their relationship(s) with the people to whom they're speaking. For example, at a company dinner you'd speak differently to coworkers with whom you work everyday than with someone from another department than with your boss than with another section head, than with the head of your company. In such a situation it would not be unlikely that you'd make use of all four categories of formality (perhaps not casual, but again that depends on your relationships with people). Japanese has a salient saying for this, 「空気｛くうき｝を読{よ｝む」which broadly means to "read the situation in order to decide how to conduct oneself." If your goal is to become an "expert," then it's important to learn not only the different formality levels, but also to be able to distinguish when it's prudent to use which.
