Remove ivy first from trees or shrubs to create a 3- to 6-foot ring or "lifesaver" around each major plant's base and from fences where it is unwanted. Then remove ivy from the ground, pulling and digging up as much root as possible to prevent re-climbing.
Cut thick ivy "trunks" first, making cuts at ground level and as high as you can reach. Use hand pruners, loppers and a pruning saw, as needed. Starting at the top, peel back each ivy trunk or stem. Use a pry bar, gently, if necessary.
Pull ivy vines and roots back to a distance of at least 3 to 6 feet. Keep going after the roots, using the shovel as needed to help remove major root sections.
Mulch the bare soil with 5 or 6 inches of wood chips or similar mulch to conserve soil moisture and protect the soil, keeping the mulch at least 3 inches from tree or shrub trunks to prevent disease or pest transmission.
Pile up ivy roots and vines in a hot, sunny spot and let them sit, dry out and die, and start to decompose. Dispose of dried ivy as yard waste.
Pull up any "missed ivy" the next year; this will take relatively little time and effort. Repeat the process again the next year to get the last of it.
Apply glyphosate to eradicate remaining ivy, if desired, only in the second and third year of ivy control. Apply glyphosate to ivy in very early spring when plants have two to four just-expanded leaves; delaying even a few weeks limits its effectiveness. Follow the label instructions for application frequency and dosage.