Pictures of the weeds plus your location and zone would be immensely helpful. Different types of weeds take different treatments in different locals. Also, a soil test at your local extension office must be done before you can realistically know what to do for the soil.
hortster

Many "weeds" are wild flowers.
The quickest way to "eliminate: these "weeds" is to spray a plant poison, but that is also the least environmentally friendly way to do that.
Tilling can work and that also works to allow you to amend the soil as a start toward a good, healthy soil.
However, as hortster stated starting with a good, reliable soil test, so you know your soils pH (and what to do to correct that) is as well as the nutrients in the soil and the balance of those nutrients. These simple soil tests might also be of some help.
1) Soil test for organic matter. 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. For example, 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 drainsâ 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.

Hand pull the weeds before they seed. That will help eliminate them for next year. Once they seed, they'll be back!

Also know what you are dealing with. Some are perennial and will come back yearly from their roots.

Some spread by suckering up from roots. You definitely need Round-up or one of it's competitors for those!

Good luck with your adventure!

Suzi