Think you have poison oak. Do a search but what I would guess it is. Good luck. Be ready for a long battle. I use 2-4D and/or Crossbow (kills shrubs broadleafs but won't harm grass) while internet suggest also using glyphosate (Round up). I always suggest buying straight and dilluting it yourself. Add sme liquid detergent as well as something to lower pH to less than 7.0. Basic water ties up Glysophate.
Internet says dig the roots up, yeah, good luck with that as they are everywhere. Small broken piece will regerm next year.
My choice would be to use glyphosate just like seed says. Don't spare the stuff, and give it a chance to work.
It won't take too long and you'll start to make some headway.
I always wonder if using round-up and triclopyr together would be self-defeating. Does the round-up kill the leaf so quickly that the leaf can't properly take up the triclopyr and send it through the entire plant?
Would it be more effective to spray first with triclopyr then a couple days later spray with round-up? I don't know. There are products with the two chemicals together. Maybe Crossbow is one.
Which ever way you choose, a squirt of dish detergent in with the water/spray will help the chemical adhere to the leaf.
I can't come up with anything that looks like that an causes instant pain, either from nettles, thorns, or sap. I kind of go with So Lucky, was some other plant growing in there that you touched? Or did you touch a stinging caterpillar? We all have different allergies and reactions so it's kind of hard to judge just by your reaction, two days of pain, but I've had some really instantaneous pain with stinging caterpillars.
My suggestion is you very carefully take a piece of that to your county extension agent and get them to identify it. Include bark, leaves, any flowers or seed pods. If they can't find anyone local to identify it they should send it to Urbana-Champaign. Identifying plants is something the extension agent should be able to manage. They should also have a recommended way to control it.
Don't know if you just put your hand on a bee or other stinging insect, but to me those look like honeysuckle vines....I've got some of the same looking stuff growing up my porch and garden arbor. It tends to have air roots like you show, especially when well established.
This is a variegated Euonymus I have in my front yard, also invasive. I have to prune it several times a year to keep it under control in a shrub shape. Compare with the leaves on yours. It sends out long stems with aerial roots and forces its way under the siding if I don't keep up after it. It was here when we moved in, so its days are numbered when I can afford some nice evergreens for my landscape.
The last image stumps me. The leaves look a lot longer/thinner than the other and radiate out. It doesn't match up with poison oak/ivy/sumac. They wouldn't sting immediately, but would cause an itchy rash days later. I don't believe it is a wild plant. Could be something exotic. She probably planted them both on purpose. Maybe she had neighbors like @Smart Red 's! That would certainly keep them out, lol.
If you cut it out as much as possible and paint straight concentrated roundup or crossbow on the stem, that should kill it. May take a couple applications.
i think Journey is right in her ID of one of those plants. i've been told it is invasive too but my parents have it & so far it has been kept in check in their yard. it might just be the cold weather we have here that helps do that.
as for the stinging plant i don't have much of a clue. i have a plant here that if you go to pull it & you grab the wrong area you could get touched by the little sharp nubs by the leaves, it's oriental bittersweet. it's a nasty vine that chokes out anything it comes in contact with. it will even compete with strangling itself.