How did Shakespeare's plays get handed down through the generations?

Question

Also what was the time period called when Elizabeth 1 and James 1 reigned?

Answer 1

Shakespeare's plays were written down in "quartos" and then transferred to the First Folio, which was published I think some 10 years after his death in 1616. From then on, he has remained published and his works therefore come down to us today.

The time periods you want are Tudor (or Elizabethan) and Stuart.

Answer 2

They were written and stored in libraries (they had libraries in England at the time)