Are those by any chance crab apple trees?
In the second picture, you can see the 3 leaves, and berries; it is definitely poison ivy.
I can't see the leaves well enough to determine varieties of plants but I can tell you some information. You have 3 different issues going on.
1) There are numerous root suckers coming up from the roots of the tree that look like long straight tree branches that grow very fast and straight up.
2) You have water sprouts coming off the main trunk of the trees that are similar to suckers.
3) You have some type of vine growing up the trunks of the trees. These can be identified by the aerial rootlets that you see coming off the vine. It is hard to tell what this vine is as I can't see the leaves but it could be poison ivy/oak.
In all three cases the answer is to cut them off flush to the ground/trunk. If indeed that is poison ivy, than you will want to be careful cutting by wearing rubber gloves and avoiding contact with the sawdust. Treat the fresh cut stumps of the vine with Tordon to prevent regrowth. DO NOT put Tordon on the root suckers or the water sprouts as it will kill the trees.
I believe there is a crabapple tree and a couple mulberry trees. There are definitely a ton of suckered and at least a little bit of poison ivy. The trees are in our yard so I will be carefully trying to clean them up a bit. Thanks for all the help guys.