Not an Elizabethan either, but I'm a former medievalist with an Elizabethan ex-boyfriend who's steeped in the stuff. Walter Raleigh, Thomas Campion, Thomas Nashe, Donne for sure, Philip Sidney barely fits in (he died in 1586), but he would have been well known then, Spenser absolutely. Not much difference between poets & playwrights at this time: all the plays were in verse, but Marlowe definitely wrote poetry too. Latin was required and in fact the lingua franca of scholars and gentlemen (which is why Chaucer, Shakespeare, Dante & the Pearl Poet were such anomalies for writing in the vernacular), Greek nearly as mandatory, especially if you were studying for clerical orders, the naughty bits of Catullus were well-known, well-loved, and oft-repeated. Ovid was a favorite too. Pretty much all the Latin and Greek poets we have translations of now existed and were well known in the Late Middle Ages & Renaissance.

Hope that helps!