Does cooking break down both fibers in food vegetables and if so is are some cooking methods better for this purpose?

Visible "fibers" in food are structures composed of many, many cells. These structures can sometimes be softened, usually most effectively by low, slow cooking (e.g. braising), and sometimes not. For example, collard greens can be tough and a bit stringy, but will soften quite a bit when braised. On the other hand, long pieces of celery can stay stringy even after long cooking, so the main way to avoid unpleasant long strings is just to cut it into shorter pieces. This variability is because there are actually many different plant structures which you might consider to be a "fiber", with different compositions, i.e. different things that might need to be affected in order to disrupt the structure and soften it.

Those visible "fibers" are a completely different thing from dietary fiber (the macronutrient), which is a large family of molecules that's contained in the cells that make up the plant. There is sometimes a fair bit of dietary fiber in the visible "fibers", for example cellulose is a form of insoluble dietary fiber, and it's a structural component of plant cell walls (so plant that need to be stronger likely have more). Broadly, cooking isn't going to do a whole lot to dietary fiber molecules. It may loosen connections between cells, or even break down the cells a bit, but the dietary fiber molecules aren't going to be substantially affected.

As noted, there are several types of "fiber" in vegetable foods. Their composition is rather variable, depending on the vegetable.

The following is a very short and simplified overview of the most important factors.

The hard visible fibers like you see in celery are basically cellulose, and won't degrade or soften much (if at all) during cooking (e.g. the outer part of broccoli stems). Most vegetables don't contain much cellulose (there is always some), it's mostly present in the surface layers (which have to protect the interior) and in the visible fibers.

Kale is a bit special, in that it has a particularly thick surface, which also contains some other insoluble (wax-like) components. As neither the cellulose nor the waxes are degraded during cooking, the kale will stay rather firm (the links between surface cells will be weakened a bit, as described below).

Then, there are different processes that do soften vegetables.

In potatoes, an important part of the softening is caused by a gelatinization of the starch, which swells and disrupts the cell walls. In addition, the pectins in the cell walls are dissolved and degraded as decribed in the next paragraph.

In most other vegetables, the pectins in the cell walls partially dissolve and can be broken down. This makes the links between individual cells weaker and allows them to separate (as in carrots and such). In addition, the cells become easier to break and liberate some of the water/juice in the cells (very noticable in e.g. spinach and lettuce). Al of this makes the vegetable softer.

There are some other soluble fibers as well, but they don't play much of a role in the softening process.