Step 1: Draw a detailed world map. A hex map is an abstraction, and the hex system is a process of layered hexagonal abstraction. Having a detailed map to guide you as you build your world at various levels of magnification is necessary.
Step 2: Create a hex system. This is just a system of scales for your various scale maps. For example, a Township map could be a hexagon 6 hexes wide, with each hex representing 1 mile. A provincial map could be composed of 10 Township hex-maps. A kingdom scale map could be built from 10 provincial hex-maps, etc.
I'm using a hex system someone else invented. They based it on the rules for 5th edition, taken from the DMG, but added a scale so you could make unbelievably huge worlds. That suited my needs, and the printouts were really high quality.
At the world scale, transcribe your world map into the highest degree of abstraction. Because world maps are usually quite blocky, the hex version will likely lose some of the detail. This is why we need our detailed world map: it will help us restore that detail as we zoom in. You start at the highest scale of abstraction, because it requires the least work.
Step 3: Begin chunking. Draw the next scale down as a series of maps. Just transcribe the world map first. You want the next scale down to be big enough that you can draw the whole world at that scale in under 50 maps. You want the next scale to be small enough that you can just barely describe the world at world scale.
I was able to draw my world at continental scale in 28 maps. The continental hexes do not suit my standards. Since there is so little land on my world map, no single continental hex could be said to be land, so the whole map would have wound up just being water. As a result, I drew the kingdom scale hexes on the world scale map to describe everything. That's fine though, because there are thousands of kingdom scale hexes on my world map. Trying to do the whole world at that scale would be impossible, so the continental scale is still useful.
Step 4: Choose the map(s) at this scale that you are most likely to wind up playing in. Don't bother detailing any of the other continental scale maps unless you have to. They are available to you for the future when you run a campaign elsewhere in the world, or when someone travels a really long way. For now, we need to focus on prepping for play, and zoom in on our play area.
My hex system has a GARGANTUAN world, so the continental hex maps, composed of kingdom hex maps, are actually quite huge. Barring magic, there's basically no chance of anyone ever travelling across a map boundary at this scale over the course of a whole campaign, so I need to REALLY zoom in. You probably will not have this issue.
In the picture below, the page on the top-right is one of the continental maps prior to being detailed. The one on the bottom left is the same map after being given provincial scale hex detail. The three to the right are increasing scales of magnification, down to Township scale.
Step 5: At the finest level of detail, the maps you'll be tracking character movements on, put in all of your symbols. Towns, rivers, roads, terrain color coding, etc. If the next scale up is chunky enough for it to be clear, put as much of this information on that map as you can as well. (This will help you keep it consistent across adjacent maps) Draw as many of these maps as is necessary to cover the anticipated play area, plus a few half-finished maps out to a day's travel beyond those edges, just in case.
Step 6: Now that you have your play area for a couple of sessions built, it's time to make these hexes crawlable. For each generic terrain type, make a system of encounter charts to generate random events. If you want to go classic, these can all be combat encounters, but I like to have all types of adventure present in my games. I'm not taking a picture of my tables, because that would be boring, and my players could abuse the knowledge.
Step 7: Keep it organized and secure. Have an organization system. Best way is to use an alphanumeric coordinate system and sort them by scale and row, then add tabs to the pages you use as bookmarks or chapters. Make sure your maps are in a secure places that they cannot be damaged, stolen, or viewed by anyone you do not intend to see them. Players can metagame pretty crazy if they know the exact lay of the land.