What is the language they talk and how could I find information abouth it?
The language spoken in the play Romeo and Juliet is Elizabethan English (English from the time of Queen Elizabeth). Because four hundred years have passed, the vocabulary will have changed. (Think of how fast slang changes these days!). However, besides the simple difference in historical periods, remember that Shakespeare is writing in verse (poetry). That means that some parts of his writing would have seemed challenging even for the people of his day.
For information on Shakespeare's grammar, try this site:
For information on Shakespeare's language in general, try this one:
And if you just want a translation in more modern English, the enotes etext of the play has that in it.
They spoke (generalizing) Middle english. It was the transition between Old english (which sucks to learn) and Modern english (which also sucks to learn). ^-^ You could go to Wikipedia but if you wanna learn it, I just suggest reading it, and using context clues to figure out what means what.