Plant identification is difficult when foliage is still emerging and few distinctive traits exist yet for narrowing-down an ID. Your description sounds like Oriental Bittersweet, and it is a very common non-native (and invasive) vine. Wildlife like birds consumes the berries, spreading the plant around each year by depositing seeds as they travel. Given that, there is no permanent way to keep the plant from ever reappearing, but as with any weed, there are some basic steps to reduce their abundance.

Making sure any bare soil is covered with either a living groundcover or by mulch will help prevent these and other weed seeds from germinating, because it will block their access to soil moisture and nutrients. Dense planting can also deny weed seeds and seedlings enough light to sprout and survive. (Mulch also blocks light from any seeds lying on the soil surface, since some plant species require light to germinate.) Avoid disturbing the soil by digging or tilling, since this mixing of the soil can bring buried seeds up to the surface or a depth at which they can then germinate.

Existing vines needing eradication can either be manually removed or carefully spot-treated with systemic herbicide, which will kill the roots. Any chemical-specific tips that may apply to this particular plant are included in the resources linked in the page above.

Miri