My backyard is overrun by English ivy and poison ivy. I have been ripping them out by hand. The other vines like Virginia Creeper and some sort of wild grape vine are pretty, but they seem to be too aggressive as well. Is there a place for these last two (are they natives?) If so, how would they be kept in check or contained in their spreading?
Poison Ivy, Virginia Creeper and Wild Grape are native species while English Ivy is a non native invasive. English Ivy is one of the easiest plants to remove by hand. It only flowers/fruits once it climbs up a tree. The best way to remove it is to cut it near ground level and then cut it again about a foot higher. Allow the portion that is on the tree to die and then gently remove it from the tree. You can then focus on pulling the plant out by the roots from the ground.
The native plants are good for local wildlife and should be preserved if possible. However, regular trimming can keep them out of areas that you do not want them. Ultimately it is your property and you can remove them if you like. You might need to remove some individuals to obtain the landscape that you desire.