I can somewhat answer how the carbon fiber is made. I believe fishing rods are not actually graphite, but rather carbon fiber with a graphene matrix.
Graphene can be thought of as a sheet of atoms in a hexagonal arrangement, in only two dimensions, so that each atom is bonded to three others, kind of like a chain link fence with hexagonal openings. The bonds between the carbon atoms are very strong covalent bonds, the same as in diamond. So the sheet itself is very strong and stiff (like a diamond) if you pulled along the sheet, and it conducts electricity and heat. Graphite is a whole bunch of these sheets stacked together, however, the bonds between sheets are vanderwaal's bonds, and very weak. Thus the sheets slide over one another very easily. Powdered graphite, with very small sheets, is thus a very good lubricant, as well as used as pencil lead (your actually shearing off small graphite sheets when you write). In larger sheets, graphite varies in properties considerably depending on direction, weak and nonconducting in one direction, and stiff, strong and conducting along the sheets. The technical term for different properties in different directions is "anisotropic."
Carbon fiber can be thought of as a graphene sheet rolled up, so that it bonds as a tube instead of stacking layers of sheets, the result is a very strong fiber that doesn't want to bond strongly to anything else.
The way they are made starts with a polymer. The polymer is oxidized at relatively low temperatures (maybe 600 degrees), breaking the bonds between individual strands, but not within them, making each strand independent of others. These are then carbonized at extremely high heat (3500 degrees is an approximate) in a noble gas atmosphere, driving off everything but the carbon. The real science comes in on what polymers to use to get the most perfect structure, of course more refinement of the initial polymer, thus more expensive, is needed to get the perfect graphene tube structure. Also, the heat treatment matters a lot, lower heat favors carbon fibers to form, making higher strengths, but also is less successful at driving off impurities, and the material is very brittle. Higher heat results in a higher percentage of carbon, but the fibers tend to break up into sheets of graphite, losing strength. I'm sure the really good stuff uses high heat to get pure carbon, followed by some more heat treatments to promote carbon fiber over graphite sheets. I don't know the specific process, and it may be proprietary.
After this, you have a thin, very strong strand of carbon fiber. Twist a bunch of strands together and you have a carbon thread, like a small multistranded rope. The thread can then be woven into a carbon fiber cloth, the specifics of the weave I do not know, and I'm sure there's several to impart different properties. You end up with a flexible, yet very strong carbon fiber cloth. In most applications, and I don't think fishing rods are any different, you use several sheets of this cloth and "glue" them together with plastic. I'm sure there's just as much science in the plastic used.
I believe super high modulus carbon fiber cloth (known commercially as graphite) isn't that hard to make, it'd need better initial polymers and more heat treatment so it may be somewhat more expensive. But this isn't what holds back progress. The higher modulus, the more brittle the material. The magic comes in the various weaves, which are proprietary to different companies, and the different plastics. Get it all right, and it ALLOWS you to use higher modulus graphite, meaning higher strength, which allows thinner walled, lighter rods with faster actions. Using that kind of graphite without a properly matching weave or resin will just result in a rod thats very high in tensile strength, but as brittle as glass.