You will probably do best with convex knurls for axial running. The best known maker is Accu-Trak:
https://accu-trak.com/knurls_convex.html. You can also use knurls with a bevel on the edges and accomplish the same thing but you have to have the maker grind them for you that way.
I have that knurler, too. Its neat because you just close down slightly on the work piece and the tool will self-adjust on the tool post; then you just lock it down to get the correct height. It takes 3/4OD x 3/8W x 1/4ID knurls if I recall correctly. That would be the KPV series of Accu-Trak knurls. Note that they sell both Circular Pitch and Diametral Pitch knurls. The difference is that the DP knurls are used with fractional sized stock from 3/32 to 1". They use a pitch classification and there are only four pitches used: 64,96,128,160. The idea is that you don't need to calculate the diameter of the work before you knurl it. It is meant to work with nominal sized stock, which works for production situations.
CP knurls will fit any diameter and use a tooth per inch (TPI) classification. The smaller the number, the coarser the pattern. There are formulas for use with CP wheels that allow you to turn your work piece to a diameter that will make mistracking less likely. I've seen all sorts of arguments about whether you should calculate the diameter or not. Obviously, as hobby guys, many opt to calculate diameters and then turn the work to that diameter within tight tolerances, only to find that wheels often mistrack anyway.
For general work, you will usually be using male knurling wheels that come in either a straight pattern that form parallel lines like that on the edge of a coin or diagonal pattern that forms a diamond-shaped knurl. Diagonal knurls require that you use a RH and LH knurl in each set of wheels. For straight, you use two of the same pitch and there is no handedness.
I use mainly CP knurls and don't calculate diameters because quite often I am knurling something that is just the way I want it so I just clamp down on it with enough pressure so the knurls don't mistrack. In fact, Form-Rol, a maker of good knurls, suggest that you use enough initial pressure to "ram the die into the blank" on the first revolution to immediately form a pattern that the wheel will follow. They also recommend that you form the knurl about 90% of full depth; they consider going to full depth to be abusive and will shorten knurl life. So, I move the knurls onto the work so that half the wheel is in contact and then crank up the pressure until I hit somewhere near 90% of full depth. Once I get that, I lube it up and run the pattern in one go. You want to feed as fast as the pattern allows and try not to dwell. Once you reach the end, back it out at about twice the feed rate as you went in. This reduces the chances of flaking. Run a file over the top of the knurls to reduce the sharp edges, then a wire brush to remove any loose debris and you're done.
So, to be sure I answered your questions, you will be well served using convex knurls for running a long pattern down the axis of the blank; this is called axial running. You will do well with two types of knurls - straight and diagonal - that will form straight lines or diamonds. How coarse or fine is up to you. I prefer finer patterns, myself. A "medium" pattern is something in the range of 16 tpi. You will have to try it and see what looks best to you.