Hi Katie,
The quickest way to travel from Lima to
Cuzco is by taking a flight. There are many travel agencies in Lima but as you are on a tight time schedule you may consider booking the flight before arriving in Lima.
From Cuzco you can visit
Machu Picchu by train or by hiking the
Inca Trail. You should try to get your Machu Picchu trip booked as soon as possible as only 500 permits per day are issued.
Best wishes,
Mark