My first guess would be Virginia Creeper. It's a native vine that creeps along the ground and up into trees and along fences. I usually let it be, I've got worse things to deal with in the garden. In one area I planted native bushes and covered with a thick layer of mulch, Virginia Creeper quickly took over and made a nice looking ground cover, choking out the weeds.

shapiro, Thank you for the reply. No, this weed doesn't have tiny little hooks on its stems. After looking online for a while, I'm starting to think that it might be a type of carpetweed. It looks pretty similar and is starting to produce small white flowers (I'll post later).

Larissa, The 2nd photo is of a grape vine. It could have been planted, but could also be a volunteer wild grape. The plant in the first photo looks familiar, but it's been years since I've lived in the Mid-West, (Minnesota), so my memory is failing me... Mike

The poison ivy will need due care when removing it, and it will need to be bagged and discarded in the trash, it is not something to compost or otherwise use again. The rest can be mulch mowed, fairly frequently, to be kept under control. If the area is to be used as a garden then you need to learn more about the soil and what it needs. Start by contacting your state universities Cooperative Extension Service about having a good, reliable soil test doen so you know what your soils pH and current nutrient levels are and also dig in wioth these simple soil tests, 1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top. 2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up. 3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart. 4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell. 5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy. to see what else that soil might need.