How would I Find the energy in joules that would that would be required to raise the temperature of 20 g of ice at -20°C to steam at 120°C? ?

You will do 5 separate calculations and then add the results of those together.

1) heat to raise temp of ice from -20°C to 0°C:
q = m c (T2-T1)
q = 20 g (2.06 J/g°C) (0 - -20°C) = 824 J

2) heat to melt ice at 0°C:
q = 20 g (334.72 J/g) = 6694 J

3) heat to raise temp of water from 0°C to 100°C:
q = m c (T2-T1)
q = 20 g (4.184 J/g°C) (100 - 0°C) = 8368 J

4) heat to vaporize water:
q = 20 g (2259.36 J/g) = 45187 J

5) heat to raise temp of steam from 100°C to 120°C:
q = m c (T2-T1)
q = 20 g (2.02 J/g°C) (120-100) = 808 J

Total heat = 61836 J