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As the title hints, I'm wondering what the usage differences are between these. I've suggested to myself that it may just be a difference with nouns and adverbs, or maybe that 宜 is just a Japanese reading of the similar meaning that can be interchanged with saying you are 'well' or 'good'.
I'd like for someone to clear this up just so I know for other examples in the future, and what sort of context they'd be used in if they are meant for different occasions.
They are quite different words.
元気 is used to describe people (or animals etc) and their moods. It often means something like 'healthy and happy', 'doing well', etc; and when it's used to describe a personality it means something like 'positive and energetic'.
宜しい is basically the formal form of いい, meaning 'good (in general)', 'positive', 'satisfactory', 'permissible', etc. It's used to describe mostly situations and actions. (I don't think you can really use 宜しい to mean 'high quality' or 'nice' like you can いい, but I could be wrong.)
Be careful not to let your understanding of English words get in the way of your understanding of Japanese words. English 'good' can mean many things, including 'doing well', 'positive', 'satisfactory', 'high quality', even 'not in need of what you're offering'; and there's no single Japanese word that corresponds to all of these senses.
