Can you provide more information and hopefully a good picture clearly showing the weeds and vine and even flowers? The common names for weeds change from location to location, and sometimes are called different things in the same location, so we need a good description of the plant.

Generally you have several methods of control.

Mechanical where you pull or hoe out the weed, chemical where you spray it out (herbicides generally won't hurt animals at the rates used, especially in small areas, but they may take out the plants around them), and you have cultural controls where you modify the environment they are growing in to make it undesirable. (They like light?, give them shade. They like moisture, let the ground dry. Plant other plants that are more agressive or come up earlier or larger and outcompete them. They are a perennial? Plant annuals and keep the ground cultivated)

Another method people have recommended with unwanted vines is to mix up a small bowl of glyphosate at high strength (50% active ingredient diluted by equal parts water) and move one of the vines so it lays through the bowl and can absorb the glyphosate. Doesn't risk killing any other plant not in the bowl, and the vine soaks up the glyphosate.