It sounds like you have either field bindweed (Convolvulus arvensis) or hedge bindweed (Calystegia sepium). Field bindweed has leaves shaped like an arrowhead or a spade with rounded or pointed tips and the leaves of hedge bindweed are more heart-shaped with pointed tips and lobes. Both are trailing perennials in the morningglory family that spread by seed and rhizomes (creeping underground stems. Management recommendations are similar for both species.
You will want to make sure that plants are controlled before they produce flowers and seed to help reduce future emergence.
To control bindweed you want to make sure that the plants are physically removed before they produce seed to help reduce/prevent emergence in the years to come. Persistent physical removal of the aboveground plant material, rhizomes, and roots will eventually deplete the plant of resources, but it could take several years of this to achieve success.
Repeated tillage could be used to deplete the plant of resources, however this may not work for your area. Tillage can also spread the plant around as small rhizome fragments are chopped up and moved.
Planting tall, competitive plants nearby may help shade out the bindweed, as it is not very shade tolerant.
The use of landscape fabric on the soil may also be an option to shade out the bindweed, particularly if the fabric is laid and then covered with mulch. Any holes cut in the fabric for desirable plants may allow bindweed a new avenue to emerge. Complete death via light deprivation is reported to take between 3 to 5 years.
A non-selective herbicide, such as glyphosate (e.g. Roundup Weed and Grass Killer, though there are several other products available) can effective in controlling both of these weeds. Depending on the level of establishment it may take multiple applications to control the bindweed. It would again be important to treat the weeds as soon as possible to avoid seed production. If any of the bindweed re-emerges, a second application later in the summer would be warranted. This late summer to early-fall, timing is when glyphosate has the greatest efficacy on perennial weeds because the plants are moving resources to the root system.
When using products containing glyphosate there are a few important points to consider. First, as with any pesticide, remember to read and follow all labeled instructions. Second, glyphosate is a broad-spectrum herbicide, meaning it will injure or kill other plants contacted during application, so care is needed to avoid desired plants (i.e. foliage, green tissue, exposed roots, damaged bark). Third, be sure that the product you choose has only the active ingredient glyphosate or glyphosate + pelargonic acid. Products with additional active ingredients may have other unwanted effects and may delay the planting of other plants in the coming season(s).
Please let me know if you have any additional questions.