Vines can be found in Forests and Woodlands (starting biome) on Cragwood-trees occasionally, but also in the Jungle on all tall Wildwood-trees in a large amount, sometimes even on small bush-like ones - and more rarely on bush-like small Cragwood trees in other biomes, like Savannahs or Canyons.
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.
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.
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.
You can rotate Vines fully into all directions by holding R (as the default key) while holding and moving the left mouse button. The chosen rotation angle can then be "locked" too by simply typing "r" while looking at a rotated item, then all items of the same stack will be placed facing the same direction.
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 or more together. This makes Vines "renewable" resources (also by "farming" them from Leafies), 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 Wood Chests. Crafting these will then also help with unlocking several other important crafting recipes.
Vines do not catch fire, and Wood blocks with vines on them neither. So even if the leaves and wood from a Wildwood tree will all burn down, the hanging Vines, all blocks of Wildwood with Vines on them, as well as all blocks of Flowers and Beeswax will still prevail floating in the air.