How should I get rid of English ivy?
You'll need a multi-pronged strategy.
First, if it is in trees, cut it off at the base of the tree and spray the newly cut vine stem with a systemic herbicide (one with the ingredient glyphosate or triclopyr) at a high strength. Do not let the herbicide get on the tree. Also, to make it easier to see what has been removed, cut the vine off again at about shoulder height so that you can easily see a bare area about the tree base. Do not let new tendrils start going up the tree again. Keep it off.
Second, tackle the vines on the ground. In the spring, you can pull, prune out or even mow all the vines down to the the soil surface, so there are no mature leaves left and you've removed as much vine as you can. This will not kill the vine because its roots store lots of energy with which to grow new leaves. However, the roots will have to use up that energy to start new leaves. (You can also physically pull or dig up root, of course.)
At this point, you can smother the vines by covering them with several layers of newspapers, overlapping, (or cardboard) and then piling on at least 2 inches of mulch. They must not get any sunlight. It will take about a year to kill them, so if you see any tendrils creeping out, remove them. Do not allow ANY leaves to grow--they will just add energy to the roots and undo everything you've done so far.
To kill the roots (and vines) even better, add another step. After you have removed all the top growth you can, and the new leaves are starting to grow, the "baby" leaves leaves will be very tender and not have the waxy coating that mature leaves have that repels herbicides. So, young leaves are more susceptible to herbicides. At this time you can spray them with either of the herbicides mentioned above. Then smother the vines with mulch as also described above.
Be completely consistent for about a year and you'll beat the English ivy. Congratulations on taking it on. English ivy is a serious foreign invasive plant that has destroyed much natural habitat.