A foldable pocket saw, clippers, gloves, and a long-handled lopper are the basic tools. Prybars pull large English ivy vines off a tree's trunk so they can be cut. 

An electric chain saw hits big vines and bushes. The hedge trimmer soars through heavy infestations of multiflora rose and masses of small vines such as kudzu. 

Herbicide use should minimal but at times is essential. Its effective for infestations of porcelain berry seedlings and multiflora rose. A quart sprayer drips super-concentrate glyphosate on fresh cut stumps of vines and plants to kill roots. 

English ivy forms massive carpets that slowly kill trees and native undergrowth. SEE https://hgic.clemson.edu/factsheet/english-ivy-control/

Expose all the vines with a downward sweep of your pocket saw to remove leaves. Pry vines from the tree trunk. Cut vines as low to ground as feasible.

Carefully cut windows. Small amounts of concentrated Roundup (glyphosate) dripped on fresh cut vine stumps stop regrowth. Pull vines up by roots.