Vines do not only grow directly on Wood blocks, but also often hang down from the top of Wildwood-trees to the ground in the jungle in long strands.
Giant Wildwood trees are probably the best places to collect Vines hanging down from their tree tops by the dozen, and being really long too. These trees can easily be spotted on the area map in the shape of white dots in emerald green jungle biomes (with dark brown cross-like clearings), since the giant trees always carry Snow on top.
Vines can also be often obtained from common green Leafies, Night Twiggies and Night Leafies - either when killing and looting them or when harvesting from them after taming and feeding them. In this way, Vines are "infinite" resources too.
You can climb up and down on Vines much like on ladders, and you can also place Vines freely as blocks on top of each other to create natural ladders. Unfortunately you cannot attach Vines "flatly" onto blocks (for example on stone boulders) to ornate them or to make Wood blocks look like "natural" Vine-covered trees.
When growing Cragwood-trees from Saplings in a fitting biome (mind that Grassland often does not support trees), sometimes specimen with vines will grow, especially if you had planted 2x2 Saplings together. This also makes Vines "renewable" resources, albeit somehow indirectly and not every time, since not every single player-grown Cragwood tree will bear Vines, and not many units anyways.
Obtaining Vines is one of the unlocking requirements for the crafting recipe for Thatched Walls in your crafting-menu ("q" by default).
Vines can be used for a number of crafting-recipes like for Wood Swords, the very useful Processor and/or Wood Chests and so on. Crafting these will also help with unlocking several other important crafting recipes.