Reddit reviews The Prosody Handbook: A Guide to Poetic Form (Dover Books on Literature & Drama)

Well, a lot of classical poetry is based in rhyme and meter - it makes the piece easier to remember and chant (e.g. The Aeneid written in dactylic hexameter, Shakespeare writing in unrhymed iambic pentameter, Seuss writing in rhyming anapestic tetrameter, etc).

If you write in rhyme or follow a classical form, e.g. sonnet, then the writing should be consistent or true to that form, yes.

Much of 'modern' poetry is in free verse, which doesn't focus on rhyme or meter or form but instead on word choice and enjambment and metaphor.

In terms of learning new abilities, I'd recommend both http://www.amazon.com/Prosody-Handbook-Guide-Poetic-Literature/dp/048644967X/ and http://www.amazon.com/Teachers-Writers-Handbook-Poetic-Forms/dp/0915924609/ for theory.

In terms of poets, I'd recommend pretty much any of the Norton anthologies to start, and checking into poets that you like from there. A lot of people point to Bukowski as a modern master, but I find his work insufferable at best (that's just my opinion, though, as I tend towards the classical forms and writing in rhyme).

I'd also recommend using the search function on /r/poetry for recommendations. The question comes up fairly frequently.