Not all vines can climb all structures, so it's important to match the plant to the support. Some plants climb by twining around a support. Some use elogated leaf stalks or tendrils to anchor themselves. And some climb by attaching themselves with little adhesive disks. Examine your plant or look up the plant in a reference to determine how it climbs.
Twining plants, like pole beans, do best on trellises on which the slats or supports are less than 2 inches wide. Plants that climb with tendrils, like sweet peas, need even narrower supports. Plants that climb with adhesive disks, like many ivies, climb best on natural materials, such as wood or stone, rather than shiny vinyl.
That way, you won't disturb the root system. Anchor the trellis in the soil and/or against a building so it's sturdy enough to support the weight of the mature plant.
Set the plant a few inches in front of the trellis. Help the vine find the support by tying it loosely to get it started on its climb.
"Climbing" roses don't actually climb, they simply produce especially long canes that can be attached to a trellis.
Choose evergreen vines for year-round privacy. Use deciduous vines to provide shade in summer but allow the sun to warm the house in winter.
Many vegetables are available in both "bush" types and "vining" types. Bush varieties remain relatively small, so for your vertical garden, choose vining or climbing types.