I've been introduced to 着る (for things you put on like a shirt) and 履く (for things you put on like pants) in class, but is there a general way to just say something like
wear clothes? Or must the "method" of putting on be specified?
Yes, there is a general word:
So, you can say:
However, in daily conversation it is more common to say
Or, after rereading your question, it looks like you don't want to specify what you are wearing. In that case, the other answer is correct.
This might not be exactly what you're asking, but in Japanese, all the verbs for putting something on, like 着る, 履く or 身につける, are change-of-state verbal phrases, i.e. in their base form, they mean "put on", not "wear".
In order to express "wear" you have to express the resulting state, which is done by ～ている (でいる)