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.